(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "Fundamentals of Optics"

JENKINS! 



WHITE 



inn t jT-<C"\ I 



McCRAW-HUX 






FUNDAMENTALS OF OPTICS 



Francis A. Jenkins 
Harvey E. White 

PROFESSORS OF PHYSICS 
UNIVERSITY OF CALIFORNIA 



Third Edition 



New York Toronto London 
McGRAW-HILL BOOK COMPANY 
1957 



FUNDAMENTALS OF OPTICS 
Copyright © 1957 by the McGraw-Hill Book Company, Inc. 

Copyright, 1937, 1950, by the McGraw-Hill Book Company, Inc. Printed 
in the United States of America. All rights reserved. This book, or parts 
thereof, may not be reproduced in any form without permission of the publishers. 

Library of Congress Catalog Card Number 56-12535 

15 16 17 18 19 20- MAMM -7 5 43210 

32328 



VyS^l \^°\\- 



HARRJS COLLEGE 
PRESTON 



535 



3-£rrvJ 



S'XT.'m- 



SuiC- 






PREFACE 



The chief objectives in preparing this new edition have been simplification 
and modernization. Experience on the part of the authors and of the many 
other users of the book over the last two decades has shown that many 
passages and mathematical derivations were overly cumbersome, thereby 
losing the emphasis they should have had . As an example of the steps taken 
to rectify this defect, the chapter on reflection has been entirely rewritten 
in simpler form and placed ahead of the more difficult aspects of polar- 
ized light. Furthermore, by expressing frequency and wavelength in cir- 
cular measure, and by introducing the complex notation in a few places, 
it has been possible to abbreviate the derivations in wave theory to make 
room for new material. 

In any branch of physics fashions change as they are influenced by the 
development of the field as a whole. Thus, in optics the notions of wave 
packet, line width, and coherence length are given more prominence 
because of their importance in quantum mechanics. For the same rea- 
son, our students now usually learn to deal with complex quantities at 
an earlier stage, and we have felt justified in giving some examples of 
how helpful these can be. Because of the increasing use of concentric 
optics, as well as graphical methods of ray tracing, these subjects have 
been introduced in the chapters on geometrical optics. The elegant 
relationships between geometrical optics and particle mechanics, as in the 
electron microscope and quadrupole lenses, could not be developed because 
of lack of space; the instructor may wish to supplement the text in this 
direction. The same may be true of the rather too brief treatments of 
some subjects where old principles have recently come into prominence, 
as in Cerenkov radiation, the echelle grating, and multilayer films. 

A difficulty that must present itself to the authors of all textbooks at 
this level is that of avoiding the impression that the subject is a defini- 
tive, closed body of knowledge. If the student can be persuaded to read 
the original literature to any extent, this impression soon fades. To 
encourage such reading, we have inserted many references, to original 
papers as well as to books, throughout the text. An entirely new set of 
problems, representing a rather greater spread of difficulty than hereto- 
fore, is included. 

v 



VI PREFACE 

It is not possible to mention all those who have assisted us by sugges- 
tions for improvement. Specific errors or omissions have been pointed 
out by L. W. Alvarez, W. A. Bowers, J. E. Mack, W. C. Price, R. S. 
Shankland, and J. M. Stone, while H. S. Coleman, J. W. Ellis, F. S. 
Harris, Jr., R. Kingslake, C. F. J. Overhage, and R. E. Worley have each 
contributed several valuable ideas. We wish to express our gratitude to 
all of these, as well as to T. L. Jenkins, who suggested the simplification 
of certain derivations and checked the answers to many of the problems. 

Francis A. Jenkins 
Harvey E. White 



CONTENTS 

Preface v 

PART I. GEOMETRICAL OPTICS 

1. Light Rays 1 

2. Plane Surfaces 14 

3. Spherical Surfaces 28 

4. Thin Lenses 44 

5. Thick Lenses 62 

6. Spherical Mirrors 82 

7. The Effects of Stops 98 

8. Ray Tracing 119 

9. Lens Aberrations 130 

10. Optical Instruments 169 

PART II. PHYSICAL OPTICS 

11. Light Waves 191 

12. The Superposition of Waves 211 

13. Interference of Two Beams of Light 232 

14. Interference Involving Multiple Reflections 261 

15. Fraunhofer Diffraction by a Single Opening 288 

16. The Double Slit 311 

17. The Diffraction Grating 328 

18. Fresnel Diffraction 353 

19. The Velocity of Light 382 

20. The Electromagnetic Character of Light 407 

21. Sources of Light and Their Spectra 422 

22. Absorption and Scattering 446 

23. Dispersion 464 

24. The Polarization of Light 488 

25. Reflection 509 

26. Double Refraction 535 

27. Interference of Polarized Light 554 

28. Optical Activity 572 

29. Magneto-optics and Electro-optics 588 

PART III. QUANTUM OPTICS 

30. Photons 609 

Index ; . 625 

vii 



PART I 

GEOMETRICAL OPTICS 



CHAPTER 1 

LIGHT RAYS 



Optics, the study of light, is conveniently divided into three fields, each 
of which requires a markedly different method of theoretical treatment. 
These are (a) geometrical optics, which is treated by the method of light 
rays, (b) physical optics, which is concerned with the nature of light and 
involves primarily the theory of waves, and (c) quantum optics, which 
deals with the interaction of light with the atomic entities of matter and 
which for an exact treatment requires the methods of quantum mechanics. 
This book deals almost entirely with (a) and (6), although some of the 
salient features of (c) will be outlined in the last chapter. These fields 
might preferably be called macroscopic, microscopic, and atomic optics 
as giving a more specific indication of their domains of applicability. 
When it is a question of the behavior of light on a large scale, the repre- 
sentation by means of rays is almost always sufficient. 

1.1. Concept of a Ray of Light. The distinction between geometrical 
and physical optics appears at once when we attempt by means of 
diaphragms to isolate a single ray of light. In Fig. \A let S represent 






(o) (b) 

Fig. \A. Attempt to isolate a single ray of light. 

a source of light of the smallest possible size, a so-called point source. 
Such a source is commonly realized by focusing the light from the white- 
hot positive pole of a carbon arc on a metal screen pierced with a small 
hole.* If another opaque screen H provided with a much larger hole 
is now interposed between 8 and a white observing screen M [Fig. 1.4(a)], 
only the portion of the latter lying between the straight lines drawn from 

* The concentrated-arc lamp to be described in Sec. 21.2 also furnishes a very con- 
venient way of approximating a point source. 

3 




4 GEOMETRICAL OPTICS 

5 will be appreciably illuminated. This observation forms the basis for 
saying that light is propagated in straight lines called rays, since it can 
be explained by assuming that only the rays not intercepted by H reach 
the observing screen. If the hole in H is made smaller, as in (6) of the 
figure, the illuminated region shrinks correspondingly, so that one might 
hope to isolate a single ray by making it vanishingly small. Experiment 
shows, however, that at a certain width of H (a few tenths of a millimeter) 
the bright spot begins to widen again. The result of making the hole 
exceedingly small is to cause the illumination, although it is very feeble, 
to spread over a considerable region of the screen [Fig. 1/1 (c)]. 

The failure of this attempt to isolate a ray is due to the process called 
diffraction, which also accounts for a slight lack of sharpness of the edge 

of the shadow when the hole is wider. 
Diffraction is a consequence of the 
wave character of light and will be 
fully discussed in the section on phys- 
ical optics. It becomes important 
only when small-scale phenomena are 
being considered, as in the use of a fine 
hole or in the examination of the edge 

most optical instruments, however, we 
deal with fairly wide beams of light and the effects of diffraction can 
usually be neglected. The concept of light rays is then a very useful one 
because the rays show the direction of flow of energy in the light beam. 

1.2. Laws of Reflection and Refraction. These two laws were dis- 
covered experimentally long before their significance was understood, and 
together they form the basis of the whole of geometrical optics. They 
may be derived from certain general principles to be discussed later, but 
for the present we shall merely state them as experimental facts. When 
a ray of light strikes any boundary between two transparent substances 
in which the velocity of light is appreciably different, it is in general 
divided into a reflected ray and a refracted ray. In Fig. IB let I A 
represent the incident ray, and let it make the angle <£ with NA, the 
normal or perpendicular to the surface at A. <f> is called the angle of 
incidence and the plane defined by I A and NA is called the plane of 
incidence. 

The law of reflection may now be stated as follows : 

The reflected ray lies in the plane of incidence, and the angle of reflec- 
tion equals the angle of incidence. 

That is, I A, NA, and AR are all in the same plane and 

<t>" = <t> (la) 



LIGHT RAYS i 

The law of refraction, usually called Snell's law after its discoverer, 1 
states that 

The refracted ray lies in the -plane of incidence, and the sine of the angle 
of refraction bears a constant ratio to the sine of the angle of incidence. 

The second part of this law therefore requires that 

sin <f> 



sin <f>' 



= const. (16) 



If on the left side of the boundary in Fig. IB there exists a vacuum (or 
for practical purposes air), the value of the constant in Eq. 16 is called 
the index of refraction n of the medium on the right. By experimental 
measurements of the angles <t> and <£' one can determine the values of n 
for various transparent substances. Then, in the refraction at a bound- 
ary between two such substances having indices of refraction n and n', 
Snell's law may be written in the symmetrical form 

n sin <t> = n' sin <f>' (lc) 

Wherever it is feasible we shall use unprimed symbols to refer to the first 
medium and primed ones for the second. The ratio n'/n is often called 
the relative index of the second medium with respect to the first. The 
constant ratio of the sines in Eq. 16 equals this relative index. When 
the angle of incidence is fairly small, Eq. lc shows that the angle of 
refraction will also be small. Under these circumstances a very good 
approximation is obtained by setting the sines equal to the angles them- 
selves, so we obtain 

<t> n' 

— 7 = — FOR SMALL ANGLES (Id) 

1.3. Graphical Construction for Refraction. A relatively simple 
method for tracing a ray of light across a boundary separating two opti- 
cally transparent media is shown in Fig. \C. Because the principles 
involved in this construction are easily extended to complicated optical 
systems, the method is useful in the preliminary design of many optical 
instruments. 

After the line GH is drawn, representing the boundary separating the 
two media of index n and n', and the angle of incidence <£ of the incident 
ray J A is selected, the construction proceeds as follows: At one side of 

* Willebrord Snell (1591-1626) of the University of Leyden, Holland. He announced 
what is essentially this law in an unpublished paper in 1621. His geometrical con- 
struction required that the ratios of the cosecants of <j>' and <t> be constant. Descartes 
was the first to use the ratio of the sines, and the law is generally known as Descartes' 
law in France. 



6 



GEOMETRICAL OPTICS 



the drawing, and as reasonably close as possible, a line OR is drawn 
parallel to J A. With a point of origin 0, two circular arcs are drawn 
with their radii proportional to the two indices n and n', respectively. 

Through the intersection point R a line is drawn parallel to NN', 
intersecting the arc n' at P. The line OP is next drawn in, and parallel 




J/ « o- 

Fig. 16". Graphical construction for refraction at a piano surface. 

to it, through A , the refracted ray AB. The angle between the incident 
and refracted ray is called the angle of deviation and is given by 



= </>-<*>' 



(le) 



To prove that this construction follows the law of refraction, we apply 
the law of si7ies to the triangle ORP. 



OR 



OP 



sin <f>' sin (it — </>) 

Since sin (ir — </>) = sin <f>, OR = n, and OP = n', substitution gives 
directly 

n n' 



sin 0' sin <f> 



(1/) 



which is Snell's law (Eq. lc). 

1.4. Principle of Reversibility. The symmetry of Eqs. la and lc with 
respect to the primed and unprimed symbols shows at once that if a 
reflected or refracted ray be reversed in direction, it will retrace its original 
path. For a given pair of media with indices n and n' any one value of 
<f> is correlated with a corresponding value of <f>'. This will be equally 
true when the ray is reversed and <f>' becomes the angle of incidence in 
the medium of index n'; the angle of refraction will then be <f>. Since the 
reversibility holds at each reflecting or refracting surface, it holds also for 
even the most complicated light paths. This useful principle has more 
than a purely geometrical foundation, and it will be shown later that it 
follows from the application to wave motion of a corresponding principle 
in mechanics. 

1.5. Optical Path. In order to state a more general principle which 
will include both the law of reflection and that of refraction, it is con- 




LIGHT RAYS 7 

venient to have the definition of a quantity called the optical path. 
When light travels a distance d in a medium of refractive index n the 
optical path is the product nd. The physical interpretation of n, to be 
given later, shows that the optical path represents the distance in vacuum 
that the light would traverse in the same time that it goes the distance 
d in the medium. When there are several segments d\, d%, . . . of the 
light path in substances having different indices n i} n 2 , . . . , the optical 
path is found as follows: 

Optical path = [d] = nidi + n 2 d 2 -f* • • • = Y n4i (Iff) 

For example, let L in Fig. ID represent a lens of refractive index n' 

immersed in some liquid of index n. 

The optical path between two j^L 

points Q and Q' on a ray becomes, n 

in this case, 

[d] = ndi + n'di + nd z 

Here Q and Q' need not necessarily 

represent points on the object and 

. r * Fig. 1 D. Illustrating the concept of op- 

image; they are merely any two tical path and Fermat's principle, 
chosen points on an actual ray. 

One may also define an optical path in a medium of continuously vary- 
ing refractive index by replacing the summation by an integral. The 
paths of the rays are then curved, and the law of refraction loses its 
meaning. We shall now consider a principle which is applicable for any 
type of variation of n and hence contains within it the laws of reflection 
and refraction as well. 

1.6. Fermat's* Principle. A correct and complete statement of this 
principle is seldom found in textbooks, because the tendency is to cite it 
in Fermat's original form, which was incomplete. Using the concept of 
optical path, the principle should read 

The path taken by a light ray in going from one point to another through 
any set of media is such as to render its optical path equal, in the first 
approximation, to other paths closely adjacent to the actual one. 

The "other paths" must be possible ones in the sense that they may only 
undergo deviations where there are reflecting or refracting surfaces. Now 
Fermat's principle will hold for a ray whose optical path is a minimum 

* Pierre Fermat (1608-1665). French mathematician, ranked by some as the 
discoverer of differential calculus. The justification of his principle given by Fermat 
was that "nature is economical," but he was unaware of circumstances where exactly 
the reverse is true. 



8 



GEOMETRICAL OPTICS 



with respect to adjacent hypothetical paths. Fermat himself stated that 
the time required by the light to traverse the path is a minimum, and the 
optical path is a measure of this time. But there are plenty of cases in 
which the optical path is a maximum, or else neither a maximum nor a 

minimum but merely stationary (at 
a point of inflection) at the position 
of the true ray. 

Consider the case of a ray of light 
that must first pass through a point 
Q, then, after reflection from a plane 
surface, pass through a second point 
Q" (see Fig. IE). To find the real 
path, we first drop a perpendicular 
to GH and extend it an equal dis- 
tance on the other side to Q' . The 
straight line Q'Q" is drawn in and 
from its intersection B the line QB. 
The real light path is therefore QBQ" 
and, as can be seen from the sym- 
metry relations in the diagram, obeys the law of reflection. 

Consider now adjacent paths to points like A and C on the mirror 
surface close to B. Since a straight line is the shortest path between 
two points, both of the paths Q'AQ" and Q'CQ" are greater than Q'BQ". 




ABC x— 
Fig. IE. Illustrating Fermat's principle 
as it applies to reflection at a plane 
surface. 




Fig. IF. Illustrating Fermat's principle 
as it applies to an elliptical reflector. 



lb) 



W 



H 



e — +■ 

PlG. IG. Graphs of optical paths involv- 
ing reflection, and illustrating conditions 
for (a) maximum; (6) stationary; and 
(c) minimum light paths. Fermat's 
principle. 



By the above construction, and equivalent triangles, QA = Q'A, and 
QC = Q'C, so that QAQ" > QBQ" and QCQ" > QBQ". Therefore 
the real path QBQ" is a minimum. 

A graph of hypothetical paths close to the real path QBQ", as shown 
in the lower right of the diagram, indicates the meaning of a minimum, 



LIGHT RAYS 




Fig. \H. Geometry of a refracted ray 
used in illustrating Fermat's principle. 



and the flatness of the curve between -4 and C illustrates that to a first 
approximation adjacent paths are equal to the real optical path. 

Consider finally the optical properties of an ellipsoidal reflector as 
shown in Fig. IF. All rays emanating from a point source Q at one 
focus are reflected according to the law of reflection and come together 
at the other focus Q'. Furthermore all paths are equal in length. (It 
will be recalled that an ellipse can be drawn with a string of fixed length 
with its ends fastened at the foci.) Because all optical paths are equal, 
this is a stationary case as mentioned 
above. On the graph in Fig. 1G(6) 
equal path lengths are represented by 
a straight horizontal line. 

Some attention will here be devoted 
to other reflecting surfaces like (a) 
and (c) in Fig. IF. If these surfaces 
are tangent to the ellipsoid at the point 
B, the line NB is normal to all three 
surfaces and QBQ' is a real path for 
all three. Adjacent paths from Q to 
points along these mirrors, however, 
will give a minimum condition for the 
real path to and from reflector c, and a 
maximum condition for the real path to and from reflector a (see Fig. IG). 

It is readily shown mathematically that both the laws of reflection 
and refraction follow Fermat's principle. Figure \H, which represents 
the refraction of a ray at a plane surface, may be used to prove the law 
of refraction (Eq. lc). The length of the optical path between a point Q 
in the upper medium of index n, and another point Q' in the lower medium 
of index n' , passing through any point .4 on the surface, is 

[d] = nd+ n'd' {IK) 

where d and d' represent the distances QA and AQ', respectively. 

Now if we let h and h' represent perpendicular distances to the surface 
and p the total length of the x axis intercepted by these perpendiculars, 
we can invoke the Pythagorean theorem concerning right triangles and 
write 

d 2 = h 2 + (p - x) 2 d' 2 = h' 2 + x 2 
When these values of d and d' are substituted in Eq. \h, we obtain 

[d] = n[h 2 + (p - a:)*]i + n'(h' 2 + as*)* (It) 

According to Fermat's principle [d] must be a minimum or a maximum 
(or in general stationary) for the actual path. One method for finding 



10 GEOMETRICAL OPTICS 

a minimum or maximum for the optical path is to plot a graph of [d] 
against x and find at what value of x a tangent to the curve is parallel 
to the x axis (see Fig. IG). The mathematical means for doing the same 
thing is, first, to differentiate Eq. \i with respect to the variable x, thus 
obtaining an equation for the slope of the graph, and, second, to set this 
resultant equation equal to zero, thus finding the value of x for which the 
slope of the curve is zero. 

By differentiating Eq. It with respect to x and setting the result equal to 
zero, we obtain 



This gives 
or, more simply, 



p — x , 

= n 



[h 2 + (p - a;) 2 ]* (h' 2 + x 2 )» 



p — x , x 



By reference to Fig. 1/7 it will be seen that the multipliers of n and 
n' are just the sines of the corresponding angles, so that we have now 
proved Eq. lc, namely, 

n sin <j> = n' sin <f>' (lj) 

A diagram for reflected light, similar to Fig. IH, can be drawn and the 
same mathematics applied to prove the law of reflection. 

1.7. Color Dispersion. It is well known to those who have studied 
elementary physics that refraction causes a separation of white light into 
its component colors. Thus, as is shown in Fig. II, the incident ray of 
white light gives rise to refracted rays of different colors (really a con- 
tinuous spectrum) each of which has a different value of </>'. By Eq. lc 
the value of n' must therefore vary with color. It is customary in the 
exact specification of indices of refraction to use the particular colors 
corresponding to certain dark lines in the spectrum of the sun. Tables 
of these so-called Fraunhofer* lines, which are designated by the letters 
A, B, C, . . . , starting at the extreme red end, are given later in Tables 
21-11 and 23-1. The ones most commonly used are those in Fig. II. 

The angular divergence of rays F and C is a measure of the dispersion 
produced, and has been greatly exaggerated in the figure relative to the 

* Joseph Fraunhofer (1787-1826). Son of a poor Bavarian glazier, Fraunhofer 
learned glass grinding, and entered the field of optics from the practical side. His 
rare experimental skill enabled him to produce much better spectra than those of his 
predecessors and led to his study of the solar lines with which his name is now associ- 
ated. Fraunhofer was one of the first to produce diffraction gratings (Chap. 17). 



LIGHT RAYS 



1 I 



average deviation of the spectrum, which is measured by the angle 
through which ray D is bent. To take a typical case of crown glass, the 
refractive indices as given in Table 23-1 are 

n F = 1.53303 n D = 1.52704 n c = 1.52441 

Now it is readily shown from Eq. \d that for a given small angle $ the 
dispersion of the F and C rays (4>' F — <f>' c ) is proportional to 

n F - n c = 0.00862 

while the deviation of the D ray (<f> — 4>' D ) depends on n D — 1 = 0.52704 














1.5 


n^p^. 


»5p 




F' n C 

n D -\ 


LO 


i 




i 





Fig. 1/. Upon refraction white light is 
spread out into a spectrum. This is 
called dispersion. 



F DC 

Violet Blue Green Yellow Red 
Fio. 1/. A graph showing the variation 
of refractive index with color. 



and is thus more than sixty times as great. The ratio of these two quan- 
tities varies greatly for different kinds of glass and is an important char- 
acteristic of any optical substance. It is called the dispersive power and 
is defined by the equation 



1 _ n F — nc 
v nD — 1 



(Ik) 



The reciprocal of the dispersive power, designated by the Greek letter v, 
lies between 30 and 60 for most optical glasses. 

Figure \J illustrates schematically the type of variation of n with 
color that is usually encountered for optical materials. The numerator 
of Eq. Ik, which is a measure of the dispersion, is determined by the 
difference in the index at two points near the ends of the spectrum. The 
denominator, which measures the average deviation, represents the mag- 
nitude in excess of unity of an intermediate index of refraction. 

It is customary in most treatments of geometrical optics to neglect 
chromatic effects and assume, as we have in the next seven chapters, 



12 GEOMETRICAL OPTICS 

that the refractive index of each specific element of an optical instrument 
is that determined for yellow sodium D light. 

PROBLEMS 

1. A ray of light in air is incident on the polished surface of a piece of glass at an 
angle of 15°. What percentage error in the angle of refraction is made by assuming 
that the sines of angles in Snell's law can be replaced by the angles themselves? 
Assume n' = 1.520. 

2. A ray of light in air is incident at an angle of 45° on glass of index 1.560. Find the 
angle of refraction (a) graphically, and (6) by calculation using Snell's law. (c) What 
is the angle of deviation? Ans. (a) 27°. (6) 26°57'. (c) 18°3'. 

3. A straight hollow pipe exactly 1 m long is closed at either end with quartz plates 
10 mm thick. The pipe is evacuated, and the index of quartz is 1.460. (a) What is 
the optical path between the two outer quartz surfaces? (6) By how much is the 
optical path increased if the pipe is filled with a gas at 1 atm pressure if the index is 
1.000250? 

4. The points Q and Q' in Fig. \H arc at a distance h = 10 cm and h' = 10 cm, 
respectively, from the surface separating water of index n = 1.333 from glasss of index 
n' = 1.500. If the distance x is 4 cm, find the optical path [d] from Q to Q'. 

Ans. 30.83 cm. 
6. An approximate law of refraction was given by Kepler in the form 4> = 
<£'/(! — k sec 4>), where k = (n' — l)/n', n' being the relative index of refraction. 
Calculate the angle of incidence <t> for glass of index n' = 1 .600, if the angle of refrac- 
tion <£' = 30°, according to (a) Kepler's formula, and (6) Snell's law. 

6. White light in air is incident at an angle of 80° on the polished surface of a piece 
of barium flint glass. If the refractive indices for red C light and blue F light are 
1.5885 and 1.5982, respectively, what is the angular dispersion between these two 
colors? Ans. 16.4'. 

7. White light in air is incident at an angle of 89° on the smooth surface of a piece 
of crown glass. If the refractive indices for red C light and violet G' light are 1.5088 
and 1.5214, respectively, what is the angular dispersion between these two colors? 

8. A solid glass sphere 4 cm in radius has an index 1.50. Draw a straight line from 
a point Q on the surface of this sphere through its center and to a point Q" 6 cm beyond 
the sphere on the other side. Find by graphical construction, and measurement of 
paths, whether this path is a maximum or mininum. Ans. Minimum. 

9. Calculate the v values for the two following pieces of glass: (a) crown glass, 
nc = 1.6205, no = 1.6231, and n F = 1.6294; (6) flint glass, nc = 1.7230, n D = 1.7300, 
and np = 1.7478. 

10. Two plane mirrors are inclined to each other at an angle a. Applying the law 
of reflection, show that any ray whose plane of incidence is perpendicular to the line of 
intersection is deviated in the two reflections by an angle which is independent of the 
angle of incidence. Express this deviation in terms of a. Ans. 8 = 2(ir — a). 

11. A ray of light is incident normally on a glass plate 2 cm thick and of refractive 
index 1.60. If the plate is turned through an angle of 45° about an axis perpendicular 
to the ray, what is the increase in optical path? 

12. An ellipsoidal mirror has a major axis of 10 cm and a minor axis of 6 cm. The 
foci are 8 cm apart. If there is a point source of light at one focus Q, there are only 
two light rays that pass through the point Q" at the center. Draw such a reflector, 
and graphically determine whether these two paths are maxima, minima, or stationary. 

Ans. One maximum, one minimum. 



LIGHT RAYS 13 

13. A ray of light under water (index 1.333) arrives at the surface, making an angle 
of 40° with the normal. Using the graphical method, find the angle this ray makes 
with the normal after it is refracted into the air above (n = 1.00). 

14. A solid glass sphere 6 cm in diameter has an index n = 2.00. Parallel rays of 
light 1 cm apart and all in a single plane are incident on this sphere with one of them 
traversing the center. Find by calculation the points where each of these rays crosses 
the central undeviated ray. Ans. 5.91 cm, 5.63 cm, and 4.73 cm from first vertex. 

16. Show mathematically that the law of reflection follows from Fermat's principle. 



CHAPTER 2 

PLANE SURFACES 



The behavior of a beam of light upon reflection or refraction at a plane 
surface is of basic importance in geometrical optics. Its study will reveal 
several of the features that will later have to be considered in the more 
difficult case of a curved surface. Plane surfaces often occur in nature, 
for example as the cleavage surfaces of crystals or as the surfaces of 
liquids. Artificial plane surfaces are used in optical instruments to bring 
about deviations or lateral displacements of rays as well as to break light 
into its colors. The most important devices of this type are prisms, but 
before taking up this case of two surfaces inclined to each other, we must 
examine rather thoroughly what happens at a single plane surface. 





n<n' 



n>n' 




Fig. 2A. Reflection and refraction of a parallel beam: (a) External reflection; (6) Inter- 
nal reflection at an angle smaller than the critical angle; (c) Total reflection at the 
critical angle 

2.1. Parallel Beam. In a beam or pencil of parallel light, each ray 
meets the surface traveling in the same direction. Therefore any one 
ray may be taken as representative of all the others. The parallel beam 
remains parallel after reflection or refraction at a plane surface, as is 
shown in Fig. 2 A (a). Refraction causes a change in width of the beam 
which is easily seen to be in the ratio cos <f>'/cos </>, whereas the reflected 
beam remains of the same width. This will prove to be important for 

14 



PLANE SURFACES 



15 



intensity considerations (Sec. 25.2). There is also chromatic dispersion 
of the refracted beam but not of the reflected one. 

Reflection at a surface where n increases, as in Fig. 2A(a), is called 
external reflection. It is also frequently termed rare-to-dense reflection 
because the relative magnitudes of n correspond roughly (though not 
exactly) to those of the actual densities of materials. In Fig. 2 A (6) is 
shown a case of internal reflection or dense-to-rare reflection. In this 
particular case the refracted beam is narrow because 4>' is close to 90°. 

2.2. The Critical Angle and Total Reflection. We have already seen 
in Fig. 2A(a) that as light passes from one medium like air into another 




A 










ft f\ft 

// /// d 



Fig. 2B. Refraction and total reflection, (a) The critical angle is the limiting angle 
of refraction, (b) Total reflection beyond the critical angle. 



medium like glass or water the angle of refraction is always less than the 
angle of incidence. While a decrease in angle occurs for all angles of 
incidence, there exists a range of refracted angles for which no refracted 
light is possible. A diagram illustrating this principle is shown in Fig. 
2B, where for several angles of incidence, from to 90°, the corresponding 
angles of refraction are shown from 0° to <j> c , respectively. 

It will be seen that in the limiting case, where the incident rays approach 
an angle of 90° with the normal, the refracted rays approach a fixed 
angle <f> c beyond which no refracted light is possible. This particular 
angle <f> c , for which <f> = 90°, is called the critical angle. A formula for 
calculating the critical angle is obtained by substituting <j> = 90°, or 
sin <j> = 1, in SnelPs law (Eq. lc), 



so that 



n X 1 = n' sin 4> c 

sin 4> c = — 
n 



(2a) 



16 



GEOMETRICAL OPTICS 



I quantity which is always less than unity. For a common crown glass 
of index 1.520 surrounded by air sin 4> c = 0.6579, and <f> c = 41°8'. 

If we apply the principle of reversibility of light rays to Fig. 26(a), all 
incident rays will lie within a cone subtending an angle of 20 c , while the 
corresponding refracted rays will lie within a cone of 180°. For angles 
of incidence greater than </> c there can be no refracted light and every ray 
undergoes total reflection as shown in Fig. 2B(b). 

The critical angle for the boundary separating two optical media is 
defined as the smallest angle of incidence, in the medium of greater 
index, for which light is totally reflected. 

Total reflection is really total in the sense that no energy is lost upon 
reflection. In any device intended to utilize this property there will, 



Total reflection 
1^ (a) 



Porro 



Dove or inverting 
c) 




1 2 

Amici or roof Triple mirror Lummer-Brodhun 

Fig. 2C. Reflecting prisms utilizing the principle of total reflection. 

however, be small losses due to absorption in the medium and to reflec- 
tions at the surfaces where the light enters and leaves the medium. The 
commonest device of this kind is the total reflection prism, which is a glass 
prism with two angles of 45° and one of 90°. As shown in Fig. 2C(a), 
the light usually enters perpendicular to one of the shorter faces, is 
totally reflected from the hypotenuse, and leaves at right angles to the 
other short face. This deviates the rays through a right angle. Such a 
prism may also be used in two other ways which are illustrated in (6) 
and (c) of the figure. The Dove prism (c) interchanges the two rays, 
and if the prism is rotated about the direction of the light, they rotate 
around each other with twice the angular velocity of the prism. 




PLANE SURFACES 17 

Many other forms of prisms which use total reflection have been devised 
for special purposes. Two common ones are illustrated in Fig. 2C(d) 
and (e). The roof prism accomplishes the same purpose as the total 
reflection prism (a) except that it introduces an extra inversion. The 
triple mirror (e) is made by cutting off the corner of a cube by a plane 
which makes equal angles with the three faces intersecting at that corner. 
It has the useful property that any ray striking it will, after being inter- 
nally reflected at each of the three faces, be sent back parallel to its 
original direction. The Lummer-Brodhun "cube" shown in (/) is used in 
photometry to compare the illumina- 
tion of two surfaces, one of which is 
viewed by rays (2) coming directly 
through the circular region where the 
prisms are in contact, the other by 
rays (1) which are totally reflected in 
the area around this region. 

Since, in the examples shown, the 

angles of incidence can be as small as 

45°, it is essential that this shall exceed 

.,..., . , ,, ,, Fig. 2D. Refraction in the prism of a 

the critical angle in order that the p u lfrich refractometer. 

reflection be total. Supposing the 

second medium to be air (n' = 1), this requirement sets a lower limit on 

the value of the index n of the prism. By Eq. 2a we must have 

Vl = I ^ sin 450 
n n 

so that n ^ y/2 = 1.414. This condition always holds for glass and is 
even fulfilled for optical materials having low refractive indices such as 
lucite (n = 1.49) and fused quartz (n = 1.46). 

The principle of most accurate refractometers (instruments for the deter- 
mination of refractive index) is based on the measurement of the critical 
angle 4> c - In both the Pulfrich and Abbe types a convergent beam strikes 
the surface between the unknown sample, of index n, and a prism of 
known index n'. Now n' is greater than n, so the two must be inter- 
changed in Eq. 2a. The beam is so oriented that some of its rays just 
graze the surface (Fig. 2D), so that one observes in the transmitted light 
a sharp boundary between light and dark. Measurement of the angle 
at which this boundary occurs allows one to compute the value of <£ c 
and hence of n. There are important precautions that must be observed 
if the results are to be at all accurate.* 

* For a valuable description of this and other methods of determining indices of 
refraction see A. C. Hardy and F. H. Perrin, "Principles of Optics," 1st ed., pp. 359- 
364, McGraw-Hill Book Company, Inc., New York, 1932. 



18 



GEOMETRICAL OPTICS 



2.3. Reflection of Divergent Rays. When a divergent pencil of light 
is reflected at a plane surface, it remains divergent. All rays originating 
from a point Q (Fig. 22?) will after reflection appear to come from another 
point Q' symmetrically placed behind the mirror. The proof of this 
proposition follows at once from the application of the law of reflection 
(Eq. la), according to which all the angles labeled <£ in the figure must 
be equal. Under these conditions the distances Q A and A Q' along the 
line QAQ' drawn perpendicular to the surface must be equal: i.e., 

s' = s 

The point Q' is said to be a virtual image of Q, since when the eye receives 
the reflected rays they appear to come from a source at Q' but do not 




_a I S-lZlyj^ 



i 



j 




Fig. 2E. The reflection of a divergent 
pencil of light. 



Fig. IF. The refraction of a divergent 
pencil of light. 



actually pass through Q' as would be the case if it were a real image. 
In order to produce a real image a surface other than a plane one is 
required. 

2.4. Refraction of Divergent Rays. Referring to Fig. 2F, let us find 
the position of the point Q' where the lower refracted ray, when produced 
backward, crosses the perpendicular to the surface drawn through Q. 
Let QA = s, Q'A = s', and AB = h. Then 



so that 



h = s tan (f> = s' tan <£' 

. tan d> sin <6 cos d>' 

s = s ^7 = s 



(26) 



tan 0' " sin 0' cos <f> 
Now according to the law of refraction (Eq. lc) the ratio 

sin <t> 



We therefore have 



, = — = const, 
sin n 



, n' cos <6' 
s' = s — 



n cos 



(2c) 



PL AXE SURFACES 



19 



The ratio of the cosines is not constant. Instead, starting at the value 
unity for small <f>, it increases slowly at first, then more rapidly. As a 
consequence the projected rays do not intersect at any single point such 
at Q'. Furthermore they do not all intersect at any other point in space. 
2.6. Images Formed by Paraxial Rays. It is well known that when 
one looks at objects through the plane surface of a refracting medium, 
as for example in an aquarium, the objects are seen clearly. Actually 
one is seeing virtual images which are not in the true position of the 
objects. When one looks perpendicularly into water they appear closer 
to the surface in about the ratio 3 : 4, which is the ratio n'/n, since n' — 1 




Fig. 2G. The image seen by refraction at a plane surface. 

for air and n = 1.33 = 4/3 for water. This observation is readily under- 
stood when one considers that the rays entering the pupil of the eye will 
in this case make extremely small angles with the normal to the surface, 
as shown in Fig. 2G. Therefore both cosines in Eq. 2c are nearly equal 
to unity, and their ratio is even more nearly so. Hence, as long as the 
rays are restricted to ones that make very small angles with the normal 
to the refracting surface, a good virtual image is formed at the distance 
s' given by 



n' 
s' = — s 
n 



PARAXIAL RAYS 



(2d) 



Rays for which the angles are small enough so that we may set the cosines 
equal to unity, and the sines equal to the angles, are called paraxial rays. 

2.6. Plane -parallel Plate. When a single ray traverses a glass plate 
with plane surfaces that are parallel to each other, it emerges parallel to 
its original direction but with a lateral displacement d which increases 
with the angle of incidence </>. Using the notation shown in Fig. 2H(a), 
we may apply the law of refraction and some simple trigonometry to 
show that the displacement is given by 



, , ( , n cos <A 

d = t sm 4> [ 1 , ti I 

\ n' cos $'/ 



(26) 



20 



GEOMETRICAL OPTICS 



From 0° up to appreciably large angles, d is nearly proportional to <£, 
for as the ratio of the cosines becomes appreciably less than 1, causing 
the right-hand factor to increase, the sine factor drops below the angle 
itself in almost the same proportion.* 

If we now consider a divergent beam to be incident on such a plate 
[Fig. 2H(b)\ the different rays of the beam are not all incident at exactly 
the same angle <}>, and therefore they undergo slightly different lateral 




Fig. 2H. Refraction by a plane-parallel plate. 

shifts. For paraxial rays this yields a point image which is shifted toward 
the plate by a distance QiQ' 2 . By applying Eq. 2d successively for the two 
surfaces, and considering the image due to the first surface to be the object 
for the second, we find 



0^-1(1-5) 



(2/) 



When the plate is turned through an appreciable angle as in part (c) 
of Fig. 2H, the emergent beam becomes astigmatic, because the lateral 
displacements of the rays are such that their projections no longer pass 
even approximately through a point. This leads, as in the case of a 
single surface, to the formation of two virtual focal lines T and S. These 
two line images *S and T are parallel and perpendicular, respectively, to the 
plane of incidence and are called astigmatic images. 

2.7. Refraction by a Prism. In a prism the two surfaces are inclined 
at some angle a so that the deviation produced by the first surface is not 

* This principle is made use of in most of the moving-picture "film-editor" devices 
in common use today. 



PLANE SURFACES 



21 



annulled by the second but is further increased. The chromatic disper- 
sion (Sec. 1.7) is also increased, and this is usually the main function of a 
prism. First let us consider, however, the geometrical optics of the 
prism for light of a single color, i.e., for monochromatic light such as is 
obtained from a sodium arc. 

The solid ray in Fig. 2/ shows the path of a ray incident on the first 
surface at the angle <f>i. 

Its refraction at the second sur- 
face, as well as at the first surface, 
obeys Snell's law, so that in terms 
of the angles shown 



sin <t>! 
sin 4>[ 



it 



sin <f>2 
sin 2 



(20) 




Fig. 21. The geometry associated with re- 
fraction by a prism. 



The angle of deviation produced 
by the first surface is /3 = <f>i — <£(> 
and that produced by the second 
surface is y = <£ 2 — 0' 2 . The total angle of deviation 5 between the inci- 
dent and emergent rays is given by 



S =0 + 7 



(2h) 



Since NN' and MN' are perpendicular to the two prism faces, a is also 
the angle at N'. From triangle ABN' and the exterior angle a, we obtain 

a = <f>[ + 0' 2 (2i) 

Combining the above equations, we obtain 

« = j8 + 7 = 01 - 01 + 02 - #• = *1 + *2 - (# + 2 ) 

or 6 = 0! + 02 - a (2j) 

2.8. Minimum Deviation. When the total angle of deviation 5 for any 
given prism is calculated by the use of the above equations, it is found to 
vary considerably with the angle of incidence. The angles thus calcu- 
lated are in exact agreement with the experimental measurements. If, 
during the time a ray of light is refracted by a prism, the prism is rotated 
continuously in one direction about an axis (A in Fig. 21) parallel to the 
refracting edge, the angle of deviation 8 will be observed to decrease, reach 
a minimum, and then increase again as shown in Fig. 2.7. 

The smallest deviation angle is called the angle of minimum deviation, 
8 m , and occurs at that particular angle of incidence where the refracted 
ray inside the prism makes equal angles with the two prism faces (see 
Fig. 2K). In this special case 



01 — 02 01 = 02 & — y 



(2*0 






22 



GEOMETRICAL OPTICS 



To prove these angles equal, assume 4>\ does not equal <t> 2 when mini- 
mum deviation occurs. By the principle of the reversibility of light 
rays (see Sec. 1.4), there would be two different angles of incidence 
capable of giving minimum deviation. Since experimentally we find 



60 



50 



t 40 



30 



20 



\ 














- 














- 




Om 














, 




1 


1 


■ 



20 



30 



40 



70 



80 



90 



50 60 

Fig. 2J. A graph of the deviation produced by a 60° glass prism of index n' = 1.50. 
At minimum deviation 5 m = 37.2°, 0, = 48.6°, and <fn' = 30.0°. 

only one, there must be symmetry and the above equalities must hold. 

In the triangle ABC in Fig. 2K the exterior angle 8 m equals the sum 
of the opposite interior angles /3 -f- y. Similarly, for the triangle ABN', 

the exterior angle a equals the sum 

Consequently 

a = 2</>( 8 m = 2/3 

0i = <t>{ 4- j8 

Solving these three equations for 
</>', and <j}\, 




FlO. 2AT. The geometry of a light ray 
traversing a prism at minimum deviation. 



Since by Snell's law n'/n = sin 0,/sin <t>[, 

n' sin £(a + 8 m ) 



n = i(a + « m ) 



it 



sin £a 



(20 



The most accurate measurements of refractive index are made by plac- 
ing the sample in the form of a prism on the table of a spectrometer and 
measuring the angles a and 8 m , the latter for each color desired. When 






PLANE SURFACES 



23 



prisms are used in spectroscopes and spectrographs, they are always set 
as nearly as possible at minimum deviation because otherwise any slight 
divergence or convergence of the incident light would cause astigmatism 
in the image. A divergent pencil incident at any arbitrary angle on a 
prism yields two focal lines T and 5 similar to those shown in Fig. 2//(c). 
Only at minimum deviation do they merge to form a true point image. 

2.9. Thin Prisms. The equations for the prism become much simpler 
when the refracting angle a becomes small enough so that its sine, and 
also the sine of the angle of deviation 8, may be set equal to the angles 
themselves. Even at an angle of 0.1 rad, or 5.7°, the difference between 




Fig. 2L. Thin prisms, (a) The displacement x, in centimeters, at a distance of 1 
m, gives the power of the prism in diopters. (6) Risley prism of variable power, 
(c) Vector addition of prism deviations. 

the angle and its sine is less than 0.2 per cent. For prisms having a 
refracting angle of only a few degrees, we may therefore simplify Eq. 21 
by writing 



n' = 



sin |(5 m -f- a) 8 m + a 



and 



sin -5a a 

8 = {n' — \)a THIN PRISM IN AIR 



(2m) 



The subscript on 8 has been dropped because such prisms are always 
used at or near minimum deviation, and n has been dropped because it 
will be assumed that the surrounding medium is air, n = 1. 

It is customary to measure the 'power of a prism by the deflection of the 
ray in centimeters at a distance of 1 m, in which case the unit of power 
is called the prism diopter. A prism having a power of 1 prism diopter 
therefore displaces the ray on a screen 1 m away by 1 cm. In Fig. 2L(a) 
the deflection on the screen is x cm and is numerically equal to the power 
of the prism. For small values of 8 it will be seen that the power in 
prism diopters is essentially the angle of deviation 8 measured in units of 
0.01 rad, or 0.573°. 

For the barium flint glass of Table 23-1, ri D = 1.59144, and Eq. 2m 
shows that the refracting angle of a 1 -diopter prism should be 

0.573 



0.59144 



= 0.97 c 



24 



GEOMETRICAL OPTICS 



2.10. Combinations of Thin Prisms. In measuring binocular accomo- 
dation, ophthalmologists make use of a combination of two thin prisms of 
equal power which can be rotated in opposite directions in their own 
plane [Fig. 2L(b)]. Such a device, known as the Risley or Herschel 
prism, is equivalent to a single prism of variable power. When the 
prisms are parallel the power is twice that of either one, while when they 
are opposed the power is zero. To find how the power and direction of 
deviation depend on the angle between the components, we use the fact 
that the deviations add vectorially. In Fig. 2L(c) it will be seen that 
the resultant deviation 8 will in general be, from the law of cosines, 

8 = VV + 5 2 2 + 25,5 2 cos (2n) 

where is the angle between the two prisms. To find the angle 7 between 
the resultant deviation and that due to prism 1 alone (or, we may say, 
between the "equivalent" prism and prism 1) we have the relation 

82 sin /3 



tan 7 = 



(2o) 



5i + 82 cos 

Since almost always 8 X = 82, we may call the deviation by either com- 
ponent 5,, and the equations simplify to 

8 = y/28i 2 {l + cos 0) = J45, 2 cos 2 1 = 25, cos 2 (2p) 

sin /3 



and 
so that 



tan 7 = 



1 + cos - tan 2 
7 = 2 



(2g) 



2.11. Graphical Method of Ray Tracing. It is often desirable in the 
process of designing optical instruments to be able quickly to trace 




Fig. 2M. A graphical method for ray tracing through a prism, 
rays of light through the system. When prism instruments are encount- 
ered, the principles presented below are found to be extremely useful. 
Consider first a 60° prism of index n' = 1.50 surrounded by air of index 
n = 1.00. After the prism has been drawn to scale as in Fig. 2M, and 



PLANE SURFACES 



25 



the angle of incidence <f>i has been selected, the construction begins as in 
Fig. 1C. 

Line OR is drawn parallel to J A, and, with an origin at 0, the two 
circular arcs are drawn with radii proportional to n and n' . Line RP is 
drawn parallel to NN', and OP is drawn to give the direction of the 
refracted ray AB. Carrying on from the point P, a line is drawn parallel 
to MN' to intersect the arc n at Q. The line OQ then gives the correct 
direction of the final refracted ray BT. In the construction diagram at 
the left the angle RPQ is equal to the prism angle a, and the angle ROQ 
is equal to the total angle of deviation 8. 

2.12. Direct-vision Prisms. As an illustration of ray tracing through 
several prisms, consider the design of an important optical device known 
as a direct-vision prism. The pri- 
mary function of such an instru- 
ment is to produce a visible spectrum 
the central color of which emerges 
from the prism parallel to the inci- 
dent light. The simplest type of 
such a combination usually consists 
of a crown-glass prism of index n' 
and angle a' opposed to a flint- 
glass prism of index n" and angle 
a", as shown in Fig. 2N. 

The indices n' and n" chosen for 
the prisms are those for the central 
color of the spectrum, namely, for 
the sodium yellow D lines. Let us 
assume that the angle a" of the flint prism is selected and the construc- 
tion proceeds with the light emerging perpendicular to the last surface 
and the angle a' of the crown prism as the unknown. 

The flint prism is first drawn with its second face vertical. The 
horizontal line OP is next drawn, and, with a center at 0, three arcs 
are drawn with radii proportional to n, n', and n" . Through the inter- 
section at P a line is drawn perpendicular to AC intersecting n' at Q. 
The line RQ is next drawn, and normal to it the side AB of the crown 
prism. All directions and angles are now known. 

OR gives the direction of the incident ray, OQ the direction of the 
refracted ray inside the crown prism, OP the direction of the refracted 
ray inside the flint prism, and finally OR the direction of the emergent 
ray on the right. The angle a' of the crown prism is the supplement of 
angle RQP. 

If more accurate determinations of angles are required, the construc- 
tion diagram will be found useful in keeping track of the trigonometric 




n > rt n" 

Fig. 2JV. Graphical ray tracing applied 
to the design of a direct-vision prism. 



26 GEOMETRICAL OPTICS 

calculations. If the dispersion of white light by the prism combination 
is desired, the indices n' and n" for the red and violet light can be drawn 
in and new ray diagrams constructed proceeding now from left to right 
in Fig. 2N(b). These rays, however, will not emerge perpendicular to the 
last prism face. 

The principles just outlined are readily extended to additional prism 
combinations like those shown in Fig. 20.* It should be noted that 





Fig. 20. Direct-vision prisms used for producing a spectrum with its central color in 
line with the incident white light. 

the upper direct-vision prism in Fig. 20 is in principle two prisms of the 
type shown in Fig. 2N placed back to back. 



PROBLEMS 

1. A clear thick mineral oil (n = 1.573) is poured into a beaker to a depth of 6 cm. 
On top of this is poured a layer of alcohol (n = 1.450) 8 cm deep, (a) How far above 
or below its true position does a silver coin on the bottom of the beaker appear to an 
observer looking straight down? (6) What is the critical angle for the interface 
between the oil and alcohol, and from which side of the interface must the light 
approach? 

2. Using Snell's law, derive Eq. 2e for the lateral displacement of a ray which is 
incident on a plane-parallel plate at an angle <f>. 

3. Calculate the lateral displacements of a ray of light incident on a parallel plate 
at the following angles: (a) 10°, (6) 20°, (c) 30°, (d) 40°, and (e) 50°. Assume a plate 
thickness of 2 cm and an index of 1.50. Plot a graph of <f> against d, and draw a 
straight line tangent to the resultant curve at the origin. 

4. Make a graph of the variation of the image distance s' with the angle of the 
incident rays (# in Fig. 2F), using s' as the ordinate and <£ as the abscissa. Take 
the object point to be situated in air 3.0 cm from a plane surface of glass of refrac- 
tive index 1.573. Ans. Graph with s' = 4.72 cm at = 0°, and 5.96 cm at = 45°. 

6. The refractive index of a liquid is measured with a Pulfrich refractometer in 
which the prism has an index of 1.625 and a refracting angle of 80° (see Fig. 2D). The 
boundary between light and dark field is found to make an angle of 27°20' with the 
normal to the second face. Find the refractive index of the liquid. 

* See E. J. Irons, Am. J. Phys., 21, 1, 1953. 



PLANE SURFACES 27 

6. A crown-glass prism with an angle of 60° has a refractive index of 1.62. If the 
prism is used at an angle of incidence <tn = 70°, find the total angle of deviation 5 by 
(a) the graphical method, and (6) calculation, (c) Calculate the angle of minimum 
deviation for this same prism. Ans. (a) 52.3°. (o) 52°18'. (c) 48°12'. 

7. A 60° flint-glass prism has a refractive index of 1.75 for sodium yellow light. 
Graphically find (a) the angle of minimum deviation, and (b) the corresponding angle 
of incidence at the first surface, (c) Calculate answers for (a) and (6) . 

8. Two thin prisms are superimposed so that their deviations make an angle of 60° 
with each other. If the powers of the prisms are 6 prism diopters and 4 prism diopters, 
respectively, find (a) the resultant deviation in degrees, (b) the power of the resultant 
prism, and (c) the angle the resultant makes with the stronger of the two prisms. 

Ans. (a) 4°59'. (6) 8.7 D. (c) 23°25'. 

9. A 60° prism gives a minimum deviation angle of 39°20'. Find (a) the refractive 
index, and (b) the angle of incidence <tn. 

10. A 50° prism has a refractive index of 1.620. Graphically determine the angle of 
deviation for each of the following angles of incidence: (a) 35°, (6) 40°, (c) 45°, (d) 50°, 
(e) 60°, and (/) 70°. Plot a graph of 8 vs. 0, (see Fig. 2J). 

Ans. (a) 37.4°. (6) 36.5°. (c) 36.4°. (d) 36.9°. (e) 39.5°. (/) 44.0°. 

11. Two thin prisms have powers of 8 D each. At what angles should they be 
superimposed to produce powers of (a) 3.0 D, (b) 5.8 D, and (c) 12.5 D, respectively? 

12. Two facets of a diamond make an angle of 40° with each other. Find the angle 
of minimum deviation if n = 2.42. Ans. 71°43\ 

13. A direct-vision prism is to be made of two elements like the one shown in Fig. 
2N. The flint-glass prism of index 1.75 has an angle a" = 45°. Find the angle a 
for the crown-glass prism if its refractive index is 1.55. Solve graphically. 

14. Solve Prob. 13 by calculation using Snell's law. Ans. 66°54'. 

15. A direct-vision prism is to be made of two elements like the one shown in Fig. 
2AT. If the crown-glass prism has an angle a' = 70° and a refractive index of 1.52, 
what must be (a) the angle a", and (6) the refractive index of the flint prism? Solve 
graphically if the crown-glass prism is isosceles. 

16. Prove that as long as the angles of incidence and refraction are small enough so 
that the angles may be substituted for their sines, i.e., for a thin prism not too far 
from normal incidence, the deviation is independent of the angle of incidence and 
equal to (n — I) a. 

17. Show that, for any angle of incidence on a prism, 



sin j-(« + 5) _ , cos y(4>\ — <j>' 2 ) 
sin -^a cos tj-(^i — <t>i) 



and that the right-hand side reduces to n' at minimum deviation. 



CHAPTER 3 

SPHERICAL SURFACES 



Many common optical devices contain not only mirrors and prisms 
having flat polished surfaces but lenses having spherical surfaces with a 
wide range of curvatures. Such spherical surfaces in contrast with 
plane surfaces treated in the last chapter are capable of forming real 
images. 

Cross-section diagrams of several standard forms of lenses are shown 
in Fig. 3 A. The three converging, or positive, lenses, which are thicker 
at the center than at the edges, are shown as (a) equiconvex, (b) plano- 
convex, and (c) positive meniscus. The three diverging, or negative, lenses, 




Converging or positive lenses Diverging or negative lenses 

Fig. 3 A. Cross sections of common types of thin lenses. 

which are thinner at the center, are (d) equiconcave, (e) plano-concave, and 
(/) negative meniscus. Such lenses are usually made of optical glass as 
free as possible from inhomogeneities, but occasionally other transparent 
materials like quartz, fluorite, rock salt, and plastics are used. Although 
we shall see that the spherical form for the surfaces may not be the ideal 
one in a particular instance, it gives reasonably good images and is much 
the easiest to grind and polish. 

It is the purpose of this chapter to treat the behavior of refraction 
at a single spherical surface separating two media of different refractive 
indices, and then in the following chapters to show how the treatment 
can be extended to two or more surfaces in succession. These latter 
combinations form the basis for the treatment of thin lenses in Chap. 4, 
thick lenses in Chap. 5, and spherical mirrors in Chap. 6. 

28 



SPHERICAL SURFACES 



29 



3.1. Focal Points and Focal Lengths. Characteristic diagrams show- 
ing the refraction of light by convex and concave spherical surfaces 
are given in Fig. SB. Each ray in being refracted obeys Snell's law as 
given by Eq. lc. The principal axis in each diagram is a straight line 
through the center of curvature C. The point A where the axis crosses 
the surface is called the vertex. In diagram (a) rays are shown diverging 
from a point source F on the axis in the first medium, and refracted into a 




Fig. SB. Diagrams illustrating the focal points F and F' and focal lengths / and /', 
associated with a single spherical refracting surface of radius r separating two media 
of index n and n'. 

beam everywhere parallel to the axis in the second medium. Diagram 
(b) shows a beam converging in the first medium toward the point F, 
and then refracted into a parallel beam in the second medium. F in 
each of these two cases is called the primary focal point, and the distance / 
is called the primary focal length. 

In diagram (c) a parallel incident beam is refracted and brought 
to a focus at the point F', and in diagram (d) a parallel incident beam is 
refracted to diverge as if it came from the point F'. F' in each case is 
called the secondary focal point, and the distance /' is called the secondary 
focal length. 

Returning to diagrams (a) and (b) for reference, we now state that 
the primary focal point F is an axial point having the property that any 
ray coming from it, or proceeding toward it, travels parallel to the axis after 
refraction. Referring to diagrams (c) and (d), we make the similar state- 
ment that the secondary focal point F' is an axial point having the property 



30 



GEOMETRICAL OPTICS 



that any incident ray traveling parallel to the axis will, after refraction, pro- 
ceed toward, or appear to come from, F'. 

A plane perpendicular to the axis and passing through either focal 
point is called a focal plane. The significance of a focal plane is illus- 
trated for a convex surface in Fig. 
3C. Parallel incident rays making 
an angle with the axis are brought 
to a focus in the focal plane at a 
point Q'. Note that Q' is in line 
with the undeviated ray through the 
center of curvature C and that this 
is the only ray that crosses the bound- 
ary at normal incidence. 

It is important to note in Fig. 31? 
that the primary focal length /for the 
convex surface [diagram (a)] is not 
equal to the secondary focal length /' of the same surface [diagram (c)]. 
It will be shown in Sec. 3.4 (see Eq. 3e) that the ratio of the focal lengths 
/'// is equal to the ratio n'/n of the corresponding refractive indices. 




Fig. 3C. Illustrating how parallel in- 
cident rays are brought to a focus at Q' 
in the secondary focal plane of a single 
spherical surface. 



I 

J 



n 
n 



(3a) 



In optical diagrams it is common practice to show incident light rays 
traveling from left to right. A convex surface therefore is one in which 
the center of curvature C lies to the right of the vertex, while a concave 
surface is one in which C lies to the left of the vertex. 

If we apply the principle of the reversibility of light rays to the diagrams 
in Fig. 3B, we should turn each diagram end-for-end. Diagram (a), 
for example, would then become a concave surface with converging 
properties, while diagram (6) would become a convex surface with 
diverging properties. Note that we would then have the incident rays in 
the more dense medium, i.e., the medium of greater refractive index. 

3.2. Image Formation. A diagram illustrating image formation by a 
single refracting surface is given in Fig. 3D. It has been drawn for the 
case in which the first medium is air with an index n = 1 and the second 
medium is glass with an index n' = 1.60. The focal lengths / and /' 
therefore have the ratio 1:1.60 (see Eq. 3a). Experimentally it is 
observed that if the object is moved closer to the primary focal plane 
the image will be formed farther to the right away from F' and will be 
larger, i.e., magnified. If the object is moved to the left, farther away 
from F, the image will be found closer to F' and will be smaller in size. 

All rays coming from the object point Q are shown brought to a focus 



SPHERICAL SURFACES 



31 



at Q'. Rays from any other object point like M will also be brought 
to a focus at a corresponding image point like M'. This ideal condition 
never holds exactly for any actual case. Departures from it give rise 
to slight defects of the image known as aberrations. The elimination of 
aberrations is the major problem of geometrical optics and will be treated 
in detail in Chap. 9. 




Fig. 3D. All rays leaving the object point Q, and passing through the refracting sur- 
face, are brought to a focus at the image point Q'. 




Fig. ZE. All rays leaving the object point Q, and passing through the refracting surface, 
appear to be coming from the virtual image point Q'. 



If the rays considered are restricted to paraxial rays, a good image is 
formed with monochromatic light. Paraxial rays are defined as those rays 
which make very small angles with the axis and lie close to the axis throughout 
the distance from object to image (see Sec. 2.5). The formulas given in 
this chapter are to be taken as applying to images formed only by paraxial 
rays. 

3.3. Virtual Images. The image M'Q' in Fig. 3D is a real image in the 
sense that if a flat screen is located at M' a sharply defined image of the 
object M Q will be formed on the screen. Not all images, however, can 
be formed on a screen, as is illustrated in Fig. 3E. Light rays from an 



32 GEOMETRICAL OPTICS 

object point Q are shown refracted by a concave spherical surface separat- 
ing the two media of index n = 1.0 and n' = 1.50, respectively. The 
focal lengths have the ratio 1 : 1 .50. 

Since the refracted rays are diverging, they will not come to a focus 
at any point. To an observer's eye located at the right, however, such 
rays will appear to be coming from the common point Q'. In other words, 
Q' is the image point corresponding to the object point Q. Similarly 
M' is the image point corresponding to the object point M . Since the 
refracted rays do not come from Q' but only appear to do so, no image 
can be formed on a screen placed at M' . For this reason such an image 
is said to be virtual. 

3.4. Conjugate Points and Planes. The principle of the reversibility 
of light rays has the consequence that, if Q'M' in Fig. 3D were an object, 
an image would be formed at QM. Hence, if any object is placed at the 
position previously occupied by its image, it will be imaged at the posi- 
tion previously occupied by the object. The object and image are thus 
interchangeable, or conjugate. Any pair of object and image points 
such as M and M' in Fig. 3D are called conjugate points, and planes 
through these points perpendicular to the axis are called conjugate planes. 

If one is given the radius of curvature r of a spherical surface separating 
two media of index n and n', respectively, as well as the position of an 
object, there are three general methods that may be employed to deter- 
mine the position and size of the image. One is by graphical methods, a 
second is by experiment, and the third is by calculation using the formula 

n n' n' - n . , . 

- + 7 = -7- (36) 

In this equation s is the object distance and s' is the image distance. 
This equation, called the Gaussian formula for a single spherical surface, 
is derived in Sec. 3.10. 

Example: The end of a solid glass rod of index 1.50 is ground and 
polished to a hemispherical surface of radius 1 cm. A small object 
is placed in air on the axis 4 cm to the left of the vertex. Find the position 
of the image. Assume n = 1.00 for air. 

Solution: By direct substitution of the given quantities in Eq. 36 we 
obtain 

1 1.50 1.50 - 1.00 1.50 0^50 _ 1 

4 + 8' 1 8' 1 4 

from which s' = 6.0 cm. One concludes, therefore, that a real image 
is formed in the glass rod 6 cm to the right of the vertex. 

As an object M is brought closer to the primary focal point, Eq. 36 
shows that the distance of the image from the vertex, AM', becomes 



SPHERICAL SURFACES 33 

steadily greater and that in the limit when the object reaches F the 
refracted rays are parallel and the image is formed at infinity. Then 
we have s' = oo , and Eq. 36 becomes 

n n' _ n' — n 
s x r 

Since this particular object distance is called the primary focal length /, 
we may write 

n n' - n . 

J = —j- (3c) 

Similarly, if the object distance is made larger and eventually approaches 
infinity, the image distance diminishes and becomes equal to /' in the 
limit, s = * . Then 

n n' _ n' — n 
oo s' r 

or, since this value of s' represents the secondary focal length /', 

n' n' — n 



r r 

Equating the left-hand members of Eqs. 3c and 3d, we obtain 



(3d) 



n n' n' f fo . 

j- r or --j (3e) 

When (n' — n)/r in Eq. 36 is replaced by n/f or by n'/f according to 
Eqs. 3c and 3d, there results 

n n' n n n' n' ,_-. 

1 + 7 = 7 or - s + 7 = f (3/) 

Both these equations give the conjugate distances for a single spherical 
surface. 

3.5. Convention of Signs. The following set of sign conventions will 
be adhered to throughout the following chapters on geometrical optics, 
and it would be well to have them firmly in mind: 

1 . All figures are drawn with the light traveling from left to right. 

2. All object distances (s) are considered as positive when they are meas- 
ured to the left of the vertex, and negative when they are measured to the 
right. 

3. All image distances (s') are positive when they are measured to the right 
of the vertex, and negative when to the left. 

4- Both focal lengths are positive for a converging system, and negative 
for a diverging system. 



34 GEOMETRICAL OPTICS 

5. Object and image dimensions are positive when measured upward from 
the axis and negative when measured downward. 

6. All convex surfaces encountered are taken as having a positive radius, 
and all concave surfaces encountered are taken as having a negative 
radius. 

Example: A concave surface with a radius of 4 cm separates two media 
of refractive index n = 1.00 and n' = 1.50. An object is located in the 
first medium at a distance of 10 cm from the vertex. Find (a) the pri- 
mary focal length, (b) the secondary focal length, and (c) the image 
distance. 

Solution: To find (a), we use Eq. 3c directly to obtain 

1.0 1.5-1.0 . -4.0 cn 

T = ^— r - or /--o^-- -8-0 cm 

To find (6), we use Eq. 3d directly and obtain 

1.5 1.5-1.0 ,, -6.0 10A 

T = — -j- or /'=—= -12.0 cm 

Note that in this problem both focal lengths are negative and that 
the ratio ///' is 1/1.5 as required by Eq. 3a. The minus signs indicate 
a diverging system similar to Fig. SE. 

To find the answer to part (c), we use Eq. 3/ and obtain, by direct 
substitution, 

1.0 . 1.5 1.0 

10 + ~V = ^80 glVmg = Cm 

The image is located 6.66 cm from the vertex A, and the minus sign 
shows it is to the left of A and therefore virtual as shown in Fig. SE. 

3.6. Graphical Constructions. The Parallel-ray Method. It would 
be well to point out here that, although the above formulas hold for all 
possible object and image distances, they apply only to images formed 
by paraxial rays. For such rays the refraction occurs at or very near 
the vertex of the spherical surface, so that the correct geometrical rela- 
tions are obtained in graphical solutions by drawing all rays as though 
they were refracted at the plane through the vertex A and normal to the 
axis. 

The parallel-ray method of construction is illustrated in Figs. 'SF and 
SG for convex and concave surfaces, respectively. Consider the light 
emitted from the highest point Q of the object in Fig. SF. Of the rays 
emanating from this point in different directions the one, QT, traveling 
parallel to the axis, will by definition of the focal point be refracted to 
pass through F'. The ray QC passing through the center of curvature is 
undeviated because it crosses the boundary perpendicular to the surface. 



SPHERICAL SURFACES 



35 



These two rays are sufficient to locate the tip of the image at Q', and 
the rest of the image lies in the conjugate plane through this point. All 
other paraxial rays from Q, refracted by the surface, will also be brought 
to a focus Q'. As a check we note that the ray QS, which passes through 
the point F, will, by definition of the primary focal point, be refracted 
parallel to the axis and will cross the others at Q' as shown in the figure. 



#4 #" f » ♦ S t *, 




Fia. 3F. Parallel-ray method for graphically locating the image formed by a single 
spherical surface. 




Fig. 3G. Illustrating the parallel-ray method applied to a concave spherical surface 
having diverging properties. 

This method is called the parallel-ray method. The numbers 1, 2, 3, 
etc., indicate the order in which the lines are customarily drawn. 

When the method just described is applied to a diverging system as 
shown in Fig. 3G, similar procedures are carried out. Ray QT, drawn 
parallel to the axis, is refracted as if it came from F' . Ray QS, directed 
toward F, is refracted parallel to the axis. Finally ray QW, passing 
through C, goes on undeviated. Extending all these refracted rays 
back to the left finds them intersecting at the common point Q'. Q'M' 
is therefore the image of the object QM. Note that Q'M' is not a real 
image since it cannot be formed on a screen. 

In both these figures the medium to the right of the spherical surface 
has the greater index; i.e., we have made n' > n. If in Fig. 3F the 



36 



GEOMETRICAL OPTICS 



medium on the left were to have the greater index, so that ri < n, the 
surface would have a diverging effect and each of the focal points F and 
F' would he on the opposite side of the vertex from that shown, just as 
they do in Fig. 3G. Similarly, if we made n' < n in Fig. 3G, the surface 
would have a converging effect and the focal points would he as they do in 
Fig. 3F. 

Since any ray through the center of curvature is undeviated and has 
all the properties of the principal axis it may be called an auxiliary axis. 

3.7. Oblique-ray Methods. Method 1. In more complicated optical 
systems that are treated in the following chapters it is convenient to be 
able graphically to trace a ray across a spherical boundary for any given 

W 




Fig. ZH. Illustrating the oblique-ray method for graphically locating images formed 
by a single spherical surface. 

angle of incidence. The oblique- ray methods permit this to be done with 
considerable ease. In these constructions one is free to choose any two 
rays coming from a common object point and, after tracing them through 
the system, find where they finally intersect. This intersection is then 
the image point. 

Let MT in Fig. 3H represent any ray incident on the surface from the 
left. Through the center of curvature C a dashed line RC is drawn, 
parallel to MT, and extended to the point where it crosses the secondary 
focal plane. The line TX is then drawn as the refracted ray and extended 
to the point where it crosses the axis at M '. Since the axis may here be 
considered as a second ray of light, M represents an axial object point 
and M' its conjugate image point. 

The principle involved in this construction is the following: If M T 
and RA were parallel incident rays of light, they would, after refraction, 
and by the definition of focal planes, intersect the secondary focal plane 
WF' at X. Since RA is directed toward C, the refracted ray ACX 
remains undeviated from its original direction. 

Method 2. This method is shown in Fig. 31. After drawing the axis 
MM' and the arc representing the spherical surface with a center C, 
any line such as 1 is drawn to represent any oblique ray of light. Next, 
an auxiliary diagram is started by drawing XZ parallel to the axis. 



SPHERICAL SURFACES 



37 



With an origin at 0, line intervals OK and OL are laid off proportional 
to n and n' , respectively, and perpendiculars are drawn through K, L, and 
A. From here the construction proceeds in the order of the numbers 1, 2, 
3, 4, 5, and 6. Line 2 is drawn through parallel to line 1, line 4 is 
drawn through J parallel to line 3, and line 6 is drawn through T parallel 
to line 5. 




Fig. 3/. Illustrating the auxiliary-diagram method for graphically locating images 
formed by paraxial rays. 

A proof for this construction is readily obtained by writing down 
proportionalities from three pairs of similar triangles in the two diagrams. 
These proportionalities are 



h 



i+3 
n' — n 



We now transpose n and n' to the left in all three equations. 



hn 



= i 



hn' 



= J 



h(n' — n) 



= i + j 



We finally add the first two equations and for the right-hand side substi- 
tute the third equality. 



hn , hn' 



= i + j 



and 



— 4- _ 

s 7" 



n — n 



It should be noted that to employ method 1 the secondary focal length /' 
must be known or it must first be calculated from the known radius of 
curvature and the refractive indices n and n'. Method 2 can be applied 
without knowing either of the focal lengths. 

3.8. Magnification. In any optical system the ratio between the trans- 
verse dimension of the final image and the corresponding dimension of the 
original object is called the lateral magnification. To determine the 
relative size of the image formed by a single spherical surface, reference 



38 



GEOMETRICAL OPTICS 



is made to the geometry of Fig. 3F. Here the undeviated ray 5 forms 
two similar right triangles QMC and Q'M'C. 

The theorem of the proportionality of corresponding sides requires 
that 



M'Q' CM' 



or 



II 



s — r 



MQ CM y s + r 

We will now define y'/y as the lateral magnification m and obtain 

U 



m = — = — 

y 



8 — r 

s + r 



(30) 



If m is positive, the image will be virtual and erect, while if it is nega- 
tive, the image is real and inverted. 

3.9. Reduced Vergence. In the formulas for a single spherical refract- 
ing surface (Eqs. 36 to 3/), the distance s, s', r, f, and /' appear in the 




n' 



% 



?/ // A 







Fig. 3J. Illustrating the refraction of light waves at a single spherical surface. 

denominators. The reciprocals 1/s, 1/s', 1/r, 1//, and 1//' actually 
represent curvatures of which s, s', r, f, and /' are the radii. 

Reference to Fig. 3J will show that if we think of M in the left-hand 
diagram as a point source of waves their refraction by the spherical 
boundary causes them to converge toward the image point M ' . In the 
right-hand diagram plane waves are refracted so as to converge toward 
the secondary focal point F' . Note that these curved lines representing 
the crests of light waves are everywhere perpendicular to the correspond- 
ing light rays that could have been drawn from object point to image 
point. 

As the waves from M strike the vertex A, they have a radius s and 
a curvature 1/s, and as they leave A converging toward M' , they have 
a radius s' and a curvature 1/s'. Similarly the incident waves arriving 
at A in the second diagram have an infinite radius °c and a curvature of 
1/co, or zero. At the vertex where they leave the surface the radius 
of the refracted waves is equal to/' and their curvature is equal to 1//'. 

The Gaussian formulas may therefore be considered as involving the 
addition and subtraction of quantities proportional to the curvatures 



SPHERICAL SURFACES 39 

of spherical surfaces. When these curvatures, rather than radii are used, 
the formulas become simpler in form and for some purposes more con- 
venient. We therefore introduce at this point the following quantities: 

V = - V = -, K = - P = - f P = %r (3/0 

s s r f f 

The first two of these, V and V, are called reduced vergences because 
they are direct measures of the convergence and divergence of the object 
and image wave fronts, respectively. For a divergent wave from the 
object s is positive, and so is the vergence V. For a convergent wave, on 
the other hand, s is negative, and so is its vergence. For a converging 
wave front toward the image V is positive, and for a diverging wave 
front V is negative. Note that in each case the refractive index involved 
is that of the medium in which the wave front is located. 

The third quantity K is the curvature of the refracting surface (recip- 
rocal of its radius), while the fourth and fifth quantities are, according to 
Eq. 3e, equal and define the refracting power. When all distances are 
measured in meters, the reduced vergences V and V, the curvature K, and 
the power P are in units called diopters. We may think of V as the power 
of the object wave front just as it touches the refracting surface and V 
as the power of the corresponding image wave front which is tangent to 
the refracting surface. In these new terms, Eq. 36 becomes 

V + V = P (Si) 

where P = ^-^ or P = (n' - n)K (3j) 

Example: One end of a glass rod of refractive index 1.50 is ground and 
polished with a convex spherical surface of radius 10 cm. An object is 
placed in the air on the axis 40 cm to the left of the vertex. Find (a) the 
power of the surface, and (b) the position of the image. 

Solution: To find the solution to (a), we make use of Eq. 3j, substitute 
the given distance in meters, and obtain 

For the answer to part (6), we first use Eq. 3/i to find the vergence V. 

_ 1.00 

v ~ cuo = +2 - 5 D 

Direct substitution in Eq. 3t gives 

2.5 + V = 5 from which V = +2.5 D 



y 



40 GEOMETRICAL OPTICS 

To find the image distance, we have V = n'/s', so that 

n' 1.50 



8 



or 



V 2.5 

s' = GO cm 



= 0.60 m 



This answer should be verified by the student, using one of the graphical 
methods of construction drawn to a convenient scale. 

3.10. Derivation of the Gaussian Formula. The basic equation 36 
is of sufficient importance to warrant its derivation in some detail. While 
there are many ways of performing a derivation, a method involving 
oblique rays will be given here. In Fig. 3K an oblique ray from an 




Fig. 3K. Geometry for the derivation of the paraxial formula used in locating images. 



axial object point M is shown incident on the surface at an angle 0, 
and refracted at an angle </>'. The refracted ray crosses the axis at the 
image point M' . If the incident and refracted rays MT and TM' are 
paraxial, the angles <f> and <f>' will be small enough so that we may put the 
sines of the two angles equal to the angles themselves and for Snell's 
law write 

A = VL 

<f>' n 



(3*0 



Since <f> is an exterior angle of the triangle MTC and equals the sum of 
the opposite interior angles, 

<t> = a + P (3Z) 

Similarly /3 is an exterior angle of the triangle TCM', so that /3 = <f>' + y, 
and 

*' - - y (3m) 

Substituting these values of <f> and <£' in Eq. 3/c and multiplying out, we 
obtain 

n'/3 — n'y = na -\- n0 or na -\- n'y = (n' — n)/3 

For paraxial rays a, /3, and 7 are very small angles, and we may set 
a = h/s, fJ = h/r, and y = h/s'. Substituting these values in the last 



equation, we obtain 



SPHERICAL SURFACES 



h . , h , , » h 

n- + n -, = (n — n) - 

s s r 



By canceling /i throughout we obtain the desired equation, 



n n _ n — n 

J T — 

s s r 



41 



(3n) 



3.11. Nomography. The term nomograph is derived from the greek 
words nomos, meaning law, and graphein, meaning to write. In physics 




10- -5 



Fig. 3L. Nomograph for determining the object or image distance for a single spherical 
surface, or for a thin lens. 

the term applies to certain graphical representations of physical laws, 
which are intended to simplify or speed up calculations. Figure 3L 
is a nomograph relating object and image distances as given by Eq. 3/, 
namely, 

(3o) 



71 _i_ _ n 

s + 7 " = J 



Its simplicity and usefulness become apparent when it is seen that 
any straight line drawn across the figure will intersect the three lines 
at values related by the above equation. 

Example: One end of a plastic rod of index 1.5 is ground and polished 
to a radius of +2.0 cm. If an object in air is located on the axis 12.0 cm 
from the vertex, what is the image distance? 



42 GEOMETRICAL OPTICS 

Solution: By direct substitution, and by the use of Eq. 3o, we obtain 

t = ^= +12.0 and i = -^— = r ^— r = +4.0 
n 1 n n — n 1.5 — 1 

If the straight edge of a ruler is now placed on s/n = +12.0, and 
f/n = +4.0, it will intersect the third line at s'/n' = +6.0. Since 
n' — 1.5, s' is equal to 6 X 1.5, or +9.0 cm. 

A little study of this nomograph will show that it applies to all object 
and image distances, real or virtual, and to all surfaces with positive or 
negative radii of curvature. Furthermore we shall find in Chap. 4 that 
it can be applied to all thin lenses by setting n and n' equal to unity. For 
thin lenses the three axes represent s, s', and / directly, and no calculations 
are necessary. 

PROBLEMS 

1. The left end of a long glass rod of index 1.60 is ground and polished to a convex 
spherical surface of radius 3.0 cm. A small object is located in the air and on the axis 
10.0 cm from the vertex. Find (a) the primary and secondary focal lengths, (b) the 
power of the surface, (c) the image distance, and (d) the lateral magnification. 

2. Solve Prob. 1 graphically, (a) Find the image distance by either of the oblique- 
ray methods. (6) Find the relative size of the image by the parallel-ray method. 

Ans. (a) +16.0 cm. (6) -1.0. 

3. The left end of a long plastic rod of index 1.56 is ground and polished to a convex 
spherical surface of radius 2.80 cm. An object 2.0 cm high is located in the air and 
on the axis 15.0 cm from the vertex. Find (a) the primary and secondary focal 
lengths, (b) the power of the surface, (c) the image distance, and (d) the size of the 
image. 

4. Solve Prob. 3 graphically, (a) Find the image distance by either of the oblique- 
ray methods. (6) Find the size of the image by the parallel-ray method. 

Ans. (a) +11.7 cm. (6) -1.0 cm. 

6. The left end of a water trough has a transparent surface of radius —1.5 cm. A 

small object 3.0 cm high is located in air and on the axis 9.0 cm from the vertex. Find 

(a) the primary and secondary focal lengths, (b) the power of the surface, (c) the 

image distance, and (d) the size of the image. Assume water to have an index 1.333. 

6. Solve Prob. 5 graphically, (a) Find the image distance by either of the oblique- 
ray methods, (b) Find the size of the image by the parallel-ray method. 

Ans. (a) —4.0 cm. (6) +1.0 cm. 

7. The left end of a long plastic rod of index 1.56 is ground and polished to a spheri- 
cal surface of radius —2.80 cm. An object 2.0 cm high is located in the air and on the 
axis 15.0 cm from the vertex. Find (a) the primary and secondary focal lengths, (6) 
the power of the surface, (c) the image distance, and (d) the size of the image. 

8. Solve Prob. 7 graphically, (a) Find the image distance by either of the oblique- 
ray methods. (6) Find the size of the image by the parallel-ray method. 

Ans. (a) —5.85 cm. (6) +0.5 cm. 

9. The left end of a long glass rod of index 1.666 is polished to a convex surface of 
radius 1.0 cm and then submerged in water (n = 1.333). A small object 3.0 cm high is 
located in the water and on the axis 12.0 cm from the vertex. Calculate (a) the 



SPHERICAL SURFACES 43 

primary and secondary focal lengths, (6) the power of the surface, (c) the image dis- 
tance, and (d) the size of the image. 

10. Solve Prob. 9 graphically, (a) Find the image distance by either of the oblique- 
ray methods. (6) Find the size of the image by the parallel-ray method. 

Ans. (a) -f-7.5 cm. (6) —1.5 cm. 

11. A glass rod 2.8 cm long and of index 1.6 has both ends polished to spherical 
surfaces with the following radii: T\ = +2.4 cm, and r% = —2.4 cm. An object 2.0 cm 
high is located on the axis 8.0 cm from the first vertex. Find (a) the primary and 
secondary focal lengths for each of the surfaces, (6) the image distance for the first 
surface, (c) the object distance for the second surface, and (d) the final image distance 
from the second vertex. 

12. Solve Prob. 11 graphically, after first calculating the answer to (a). 

Ans. (a) +4.0 cm, +6.4 cm, +6.4 cm, +4.0 cm. (6) +12.8 cm. (c) -10.0 cm. 
(d) +2.45 cm. 

13. A parallel beam of light enters a clear plastic bead 2.0 cm in diameter and index 
1.40. At what point beyond the bead are these rays brought to a focus? 

14. Solve Prob. 13 graphically by the method illustrated in Fig. 3/. 

Ans. +0.75 cm. 

15. A glass bead of index 1.60 and radius 2.0 cm is submerged in a clear liquid of 
index 1.40. If a parallel beam of light in the liquid is allowed to enter the bead, at 
what point beyond the far side will the light be brought to a focus? 

16. Solve Prob. 15 graphically by the method illustrated in Fig. 37. Ans. +6.0 cm. 

17. A hollow glass cell is made of thin glass in the form of an equiconcave lens. 
The radii of the two surfaces are 2.0 cm, and the distance between the two vertices is 
2.0 cm. When sealed airtight, this cell is submerged in water of index 1.333. Calcu- 
late (a) the focal lengths of each surface, and (6) the power of each surface. 

18. A spherical surface with a radius of +2.5 cm is polished on the end of a glass 
rod of index 1.650. Find its power when placed in (a) air, (6) water of index 1.333, 
(c) oil of index 1.650, and (rf) an organic liquid of index 1.820. 

Ans. (a) +26.0 D. (6) +12.7 D. (c) D. (d) -6.8 D. 



CHAPTER 4 

THIN LENSES 



Diagrams of several standard forms of thin lenses are shown in Fig. 3 A 
at the beginning of the last chapter. They are shown there as illustra- 
tions of the fact that most lenses have surfaces that are spherical in form. 
Some surfaces are convex, others are concave, and still others are plane. 
When light passes through any lens, refraction at each of its surfaces 
contributes to its image-forming properties according to the principles 
put forward in Chap. 3. Not only does each individual surface have its 
own primary and secondary focal points and planes, but the lens as a 
whole has its own pair of focal points and focal planes. 

A thin lens may be defined as one whose thickness is considered small in 
comparison with the distances generally associated with its optical properties. 
Such distances are, for example, radii of curvature of the two spherical 
surfaces, primary and secondary focal lengths, and object and image 
distances. 

4.1. Focal Points and Focal Lengths. Diagrams showing the refraction 
of light by an equiconvex lens and by an equiconcave lens are given in 
Fig. 4 A . The axis in each case is a straight line through the geometrical 
center of the lens and perpendicular to the two faces at the points of 
intersection. For spherical lenses this line joins the centers of curvature 
of the two surfaces. The primary focal point F is an axial point having 
the property that any ray coming from it, or proceeding toward it, travels 
parallel to the axis after refraction. 

Every thin lens in air has two focal points, one on each side of the 
lens and equidistant from the center. This may be seen by symmetry 
in the cases of equiconvex and equiconcave lenses, but it is also true 
for other forms provided the lenses may be regarded as thin. The 
secondary focal point F' is an axial point having the property that any inci- 
dent ray traveling parallel to the axis will, after refraction, proceed toward, or 
appear to come from, F' . The two lower diagrams in Fig. 4vl are given 
for the purpose of illustrating this definition. In analogy to the case 
of a single spherical surface (see Chap. 3), a plane perpendicular to the 
axis and passing through a focal point is called a focal plane. The sig- 
nificance of the focal plane is illustrated for a converging lens in Fig. 4B. 

44 



THIN LENSES 



45 



Parallel incident rays making an angle with the axis are brought to a 
focus at a point Q' in line with the chief ray. The chief ray in this case 
is denned as the ray which passes through the center of the lens. 



Primary 

focal 

point 




Fig. 4A. Ray diagrams illustrating the primary and secondary focal points F and F', 
and the corresponding focal lengths / and /', of thin lenses. 

The distance between the center of a lens and either of its focal points 
is called its focal length. These distances, designated / and /', usually 
measured in centimeters or inches, have a positive sign for converging 
lenses and a negative sign for diverging lenses. It should be noted in 
Fig. 4^4 that the primary focal point 
F for a converging lens lies to the 
left of the lens, whereas for a diverg- 
ing lens it lies to the right. For a 
lens with the same medium on both 
sides, we have, by the reversibility 
of light rays, 

/ = /' 



CM' e 



i ( °U 





Focal 

plane -A 


Q' 


PP^ 


F' 


w 

H 


-r — * 


Axis 



■K-r . * ,, ,, ,.,» , Fig. 42?. Illustrating how parallel inci- 

Note carefuUy the difference be- dent rays are bnmg fc t t0 / focu8 at the 

tween a thin lens in air where the secondary focal plane of a thin lens. 

focal lengths are equal and a single 

spherical surface where the two focal lengths have the ratio of the two 

refractive indices (see Eq. 3a). 

4.2. Image Formation. When an object is placed on one side or the 

other of a converging lens, and beyond the focal plane, an image is 

formed on the opposite side (see Fig. 4C). If the object is moved closer 



46 



GEOMETRICAL OPTICS 



to the primary focal plane, the image will be formed farther away from 
the secondary focal plane and will be larger, i.e., magnified. If the object 
is moved farther away from F, the image will be formed closer to F' and 
will be smaller in size. 

In Fig. 4C all the rays coming from an object point Q are shown as 
brought to a focus Q', and the rays from another point M are brought 
to a focus at M' . Such ideal conditions and the formulas given in this 
chapter hold only for paraxial rays, i.e., rays close to and making small 
angles with the lens axis. 



Object 




Fig. 4C. Image formation by an ideal thin lens. All rays from an object point Q, which 
pass through the lens, are refracted to pass through the image point Q'. 



4.3. Conjugate Points and Planes. If the principle of the reversibility 
of light rays is applied to Fig. AC, we observe that Q'M' becomes the 
object and QM becomes its image. The object and image are therefore 
conjugate just as they are for a single spherical surface (see Sec. 3.4). 
Any pair of object and image points such as M and M' in Fig. AC are 
called conjugate points, and planes through these points perpendicular 
to the axis are called conjugate planes. 

If we know the focal length of a thin lens and the position of an object, 
there are three methods of determining the position of the image. One 
is by graphical construction, the second is by experiment, and the third 
is by use of the lens formula. 



i + 4-i 

8 W / 



(4a) 



Here s is the object distance, s' is the image distance, and / is the focal 
length, all measured to or from the center of the lens. The lens equation 
(Eq. 4a), will be derived in Sec. 4.14. We will now consider the graphical 
methods. 

4.4. The Parallel-ray Method. The parallel-ray method is illustrated 
in Fig. 4D. Consider the light emitted from the extreme point Q on 
the object. Of the rays emanating from this point in different directions 
the one QT, traveling parallel to the axis, will by definition of the focal 
point be refracted to pass through F' . The ray QA, which goes through 



THIN LENSES 



4/ 



the lens center where the faces are parallel, is undeviated and meets the 
other ray at some point Q'. These two rays are sufficient to locate the 
tip of the image at Q', and the rest of the image lies in the conjugate 
plane through this point. All other rays from Q will also be brought 
to a focus at Q'. As a check, we note that the ray QF which passes 
through the primary focal point will by definition of F be refracted parallel 
to the axis and will cross the others at Q' as shown in the figure. The 
numbers 1, 2, 3, etc., in Fig. 42) indicate the order in which the lines are 
customarily drawn. 




Fig. 41). Illustrating the parallel-ray method for graphically locating the image formed 
by a thin lens. 




Fig. 422. Illustrating the oblique-ray method for graphically locating the image formed 
by a thin lens. 



4.5. The Oblique-ray Method. Let MT in Fig. 42? represent any ray 
incident on the lens from the left. It is refracted in the direction TX 
and crosses the axis at M' . The point X is located at the intersection 
between the secondary focal plane F'W and the dashed line RR' drawn 
through the center of the lens parallel to MT. 

The order in which each step of the construction is made is again 
indicated by the numbers 1, 2, 3, ... . The principle involved in this 
method may be understood by reference to Fig. 42?. Parallel rays 
incident on the lens are always brought to a focus at the focal plane, 
the ray through the center being the only one undeviated. Therefore, 



48 GEOMETRICAL OPTICS 

if we actually have rays diverging from M, as in Fig. AE, we may find 
the direction of any one of them after it passes through the lens by making 
it intersect the parallel line RR' through A in the focal plane. This 
construction locates X and the position of the image M'. Note that RR' 
is not an actual ray in this case and is treated as such only as a means of 
locating the point X. 

4.6. Use of the Lens Formula. To illustrate the application of Eq. 4a 
to find the image position, we select an example in which all quantities 
occurring in the equation have a positive sign. Let an object be located 
6 cm in front of a positive lens of focal length +4 cm. As a first step we 
transpose Eq. 4a by solving for s'. 

»' = £-{ (46) 

Direct substitution of the given quantities in this equation gives 

The image is formed 12 cm from the lens and is real, as it will always be 
when s' has a positive sign. In this instance it is inverted, corresponding 
to the diagram in Fig. AC. These results can be readily checked by 
•either of the two graphical methods presented above. 

The sign conventions to be used for the thin-lens formulas are identical 
to those for a single spherical surface given on page 33. 

4.7. Lateral Magnification. A simple formula for the image magni- 
fication produced by a single lens may be derived from the geometry 
of Fig. AD. By construction it is seen that the right triangles QMA 
and Q'M'A are similar. Corresponding sides are therefore proportional 
to each other, so that 

M'Q' _ iM' 
MQ AM 

where AM' is the image distance s' and AM is the object distance s. 
Taking upward directions as positive, y = MQ, and y' = — Q'M' ; so we 
have by direct substitution y'/y = —s'/s. The lateral magnification 
is therefore 

m = V - = - - (4c) 

y s 

When s and s' are both positive, as in Fig. AD, the negative sign of the 
magnification signifies an inverted image. 

4.8. Virtual Images. The images formed by the converging lenses in 
Figs. AC and AD are real in that they can be made visible on a screen. 



THIN LENSES 



49 



They are characterized by the fact that rays of light are actually brought 
to a focus in the plane of the image. A virtual image cannot be formed 
on a screen (see Sec. 2.3). The rays from a given point on the object 
do not actually come together at the corresponding point in the image; 
instead they must be projected backward to find this point. Virtual 
images are produced with converging lenses when the object is placed 
between the focal point and the lens, and by diverging lenses when the 
object is in any position. Examples of these cases are shown in Figs. 
4F and AG. 




Fig. 4F. Illustrating the parallel-ray method for graphically locating the virtual image 
formed by a positive lens when the object is between the primary focal point and the 
lens. 



Figure 4F shows the parallel-ray construction for the case where a 
positive lens is used as a magnifier, or reading glass. Rays emanating 
from Q are refracted by the lens but are not sufficiently deviated to 
come to a real focus. To the observer's eye at E these rays appear to 
be coming from a point Q' on the far side of the lens. This point repre- 
sents a virtual image, because the rays do not actually pass through Q' ; 
they only appear to come from there. Here the image is right side up 
and magnified. In the construction of this figure, ray QT parallel to 
the axis is refracted through F', while ray QA through the center of the 
lens is undeviated. These two rays when extended backward intersect 
at Q'. The third ray QS, traveling outward as though it came from F, 
actually misses the lens, but if the latter were larger, the ray would be 
refracted parallel to the axis, as shown. When projected backward, it 
also intersects the other projections at Q'. 

Example: If an object is located 6 cm in front of a lens of focal length 
+ 10 cm, where will the image be formed? 



50 GEOMETRICAL OPTICS 

Solution: By making direct substitutions in Eq. 46 we obtain 



s' = 



(+6) X (+10) +60 



= -15 



(+6) - (+10) -4 

The minus sign indicates that the image lies to the left of the lens. 
Such an image is always virtual. The magnification is obtained by the 
use of Eq. 4c, 

-15 



m = = — 



+6 



= +2.5 



The positive sign means that the image is erect. 

In the case of the negative lens shown in Fig. 40? the image is virtual 
for all positions of the object, is always smaller than the object, and lies 




Fig. <&?. Illustrating the parallel-ray method for graphically locating the virtual image 
formed by a negative lens. 



closer to the lens than the object. As is seen from the diagram, rays 
diverging from the object point Q are made more divergent by the lens. 
To the observer's eye at E these rays appear to be coming from the point 
Q' on the far side of but close to the lens. In applying the lens formula 
to a diverging lens it must be remembered that the focal length / is 
negative. 

Example: An object is placed 12 cm in front of a diverging lens of 
focal length 6 cm. Find the image. 

Solution: We substitute directly in Eq. 46, to obtain 



s' = 



(+12) X (-6) _ ^72 
(+12) - (-6) +18 



THIN LENSES 51 

from which s' = — 4 cm. For the image size Eq. 4c gives 

s' -4,1 

m = ~ 8 " ~ 12 = + 3 

The image is therefore to the left of the lens, virtual, erect, and one-third 
the size of the object. 

4.9. Lens Makers' Formula. If a lens is to be ground to some speci- 
fied focal length, the refractive index of the glass must be known. It is 
customary for manufacturers of optical glass to specify the refractive 
index for yellow sodium light, the D line. Supposing the index to be 
known, the radii of curvature must be so chosen as to satisfy the equation 



} - <» - 1 > fe - s) < 4d > 



As the rays travel from left to right through a lens, all convex surfaces 
encountered are taken as having a positive radius, and all concave surfaces 
encountered, a negative radius. For an equiconvex lens like the one in 
Fig. 3 A (a), ri for the first surface is positive and r 2 for the second surface 
negative. Substituting the value of 1// from Eq. 4a, we may write 

Example: A plano-convex lens having a focal length of 25 cm [Fig. 
3.4(6)] is to be made of glass of refractive index n = 1.520. Calculate 
the radius of curvature of the grinding and polishing tools that must be 
used to make this lens. 

Solution: Since a plano-convex lens has one flat surface, the radius for 
that surface is infinite, and r x in Eq. 4d is replaced by « . The radius 
r 2 of the second surface is the unknown. Substitution of the known 
quantities in Eq. 4d gives 

i = (1.520- 1)(— --) 
25 ' \w r 2 / 

Transposing and solving for r 2 , 

1 = 0.520 (0-1)= -2^2 
2o \ r 2 / r 2 

giving r 2 = - (25 X 0.520) = - 13.0 cm 

If this lens is turned around, as in the figure, we will have r x = +13.0 cm 
and r 2 = =0 . 



52 



GEOMETRICAL OPTICS 



4.10. Thin-lens Combinations. The principles of image formation pre- 
sented in the preceding sections of this chapter are readily extended to 
optical systems involving two or more thin lenses. Consider, for example, 
two converging lenses spaced some distance apart as shown in Fig. AH. 
Here an object Q\M\ is located at a given distance s in front of the first 
lens, and an image Q' 2 M 2 is formed some unknown distance s' 2 from the 
second lens. We first apply the graphical methods to find this image 
distance and then show how to calculate it by the use of the thin-lens 
formula. 




si— A 



Fig. 4//. Illustrating the parallel-ray method for graphically locating the final image 
formed by two thin lenses. 

The first step in applying the graphical method is to disregard the 
presence of the second lens and find the image produced by the first one 
alone. In the diagram the parallel-ray method, as applied to the object 
point Qi, locates a real and inverted image at Q[. Any two of the three 
incident rays 3, 5, and 6 are sufficient for this purpose. Once Q[ is 
located, we know that all the rays leaving Qi will, upon refraction by the 
first lens, be directed toward Q[. Making use of this fact, we construct 
a fourth ray by drawing line 9 back from Q[ through A 2 to W. Line 10 
is then drawn in connecting W and Q\. 

The second step is to imagine the second lens in place and to make the 
following changes: Since ray 9 is seen to pass through the center of lens 2, 
it will emerge without deviation from its previous direction. Since ray 7 
between the lenses is parallel to the axis, it will upon refraction by the 
second lens pass through its secondary focal point F' 2 . The intersection 
of rays 9 and 1 1 locates the final image point Q' 2 . Qi and Q[ are conjugate 
points for the first lens, Q 2 and Q 2 are conjugate points for the second 
lens, while Qi and Q 2 are conjugate for the combination of lenses. When 
the image Q 2 M 2 is drawn in, corresponding pairs of conjugate points 
on the axis are Mi and M[, M 2 and M 2 , and M x and M' 2 . 

The oblique-ray method given in Fig. AE is applied to the same two 
lenses in Fig. 47. A. single ray is traced from the object point M to the 



THIN LENSES 



53 



final image point M' 2 . The lines are drawn in the order indicated. The 
dotted line 6 is drawn through Ai parallel to ray 4 to locate the point R[. 
The dotted line 9 is drawn through A 2 parallel to ray 7 to locate #2. 
This construction gives the same conjugate points along the axis. Note 
that the axis itself is considered as the second light ray in locating the 
image point M' 2 . 

By way of comparison and as a check on the graphical solutions, 
we can assign specific values to the focal lengths of the lenses and apply 
the thin-lens formula to find the image. Assume that the two lenses have 




Fig. 47. Illustrating the oblique-ray method for graphically locating the final image 
formed by two thin lenses. 



focal lengths of -f-3 and +4 cm, respectively, that they are placed 2 cm 
apart, and that the object is located 4 cm in front of the first lens. 

We begin the solution by applying Eq. 46 to the first lens alone. The 
given quantities to be substituted are Si = +4 cm and /1 = +3 cm. 

. _ «iX/i _ (+4) X (+3) _ 
Sl " sT^h " (+4) - (4-3) " + 12 Cm 

The image formed by the first lens alone is, therefore, real and 12 cm 
to the right of A x . The image becomes the object for the second lens, 
and since it is only 10 cm from A 2 , the object distance s 2 becomes — 10 cm. 
The minus sign is necessary and results from the fact that the object 
distance is measured to the right of the lens. We say that the image 
produced by the first lens becomes the object for the second lens. Since 
the rays are converging toward the image of the first lens, the object 
for the second lens is virtual and its distance therefore has a negative 
value. Applying the lens formula (Eq. 46) to the second lens, we have 
s 2 = — 10 cm and / 2 = +4 cm. 



So = 



_ (-10 ) X (+4) _ 



(-10) - (+4) 



= +2.86 cm 



The final image is 2.86 cm to the right of lens 2 and is real. 

4.11. Object Space and Image Space. For every position of the 
object there is a corresponding position for the image. Since the image 



54 GEOMETRICAL OPTICS 

may be either real or virtual and may lie on either side of the lens, the 
image space extends from infinity in one direction to infinity in the other. 
But object and image points are conjugate; so the same argument holds 
for the object space. In view of their complete overlapping, one might 
wonder how it is that the distinction between object and image space is 
made. This is done by defining everything that pertains to the rays 
before they have passed through the refracting system as belonging to 
the object space and everything that pertains to them afterward as 
belonging to the image space. Referring to Fig. AH, the object Qi and 
the rays QiT, QiA h and QiV are all in the object space for the first lens. 
Once these rays leave that lens, they are in the image space of the first 
lens, as is also the image Q[. This space is also the object space for the 
second lens. Once the rays leave the second lens, they and the image 
point Q' 2 are in the image space of the second lens. 

4.12. The Power of a Thin Lens. The concept and measurement of 
lens power correspond to those used in the treatment of reduced vergence 
and the power of a single surface as given in Sec. 3.9. The power of a 
thin lens in diopters is given by the reciprocal of the focal length in 
meters. 

P =J Di0pters = focal length, m (4/) 

For example, a lens with a focal length of +50 cm has a power of 
1/0.50 m = +2 D (P = +2.0 D), whereas one of —20 cm focal length 
has a power of 1/0.20 m = -5 D (P = -5.0 D), etc. Converging 
lenses have a positive power, while diverging lenses have a negative power. 

By making use of the lens makers' formula (Eq. 4d) we may write 



P = (n - 1) 



{k-3 



where r t and r 2 are the two radii, measured in meters, and n is the refrac- 
tive index of the glass. 

Example: The radii of both surfaces of an equiconvex lens of index 1 .60 
are equal to 8 cm. Find its power. 

Solution: The given quantities to be used in Eq. 4g are n = 1.60, 
ri = 0.08 m, and r 2 = —0.08 m (see Fig. 3 A for the shape of an equi- 
convex lens). 

p = °- 60 (oi) = +150D 



THIN LEN8ES 



55 



Spectacle lenses are made to the nearest quarter of a diopter, thereby 
reducing the number of grinding and polishing tools required in the 
optical shops. Furthermore, the sides next to the eyes are always con- 
cave to permit free movement of the eyelashes and yet to keep as close 
to and as normal to the axis of the eye as possible. Note: it is important 
to insert a plus or minus sign in front of the number specifying lens power; 
thus, P = +3.0 D, P = -4.5 D, etc. 

4.13. Thin Lenses in Contact. When two thin lenses are placed in con- 
tact as shown in Fig. 4J, the combination will act as a single lens with 



L, L 



Axis 




Fig. 4J\ The power of a combination of thin lenses in contact is equal to the sum of the 
powers of the individual lenses. 



two focal points symmetrically located at F and F' on opposite sides. 
Parallel incoming rays are shown refracted by the first lens toward its 
secondary focal point F[. Further refraction by the second lens brings 
the rays together at F'. This latter is defined as the secondary focal 
point of the combination, and its distance from the center is defined 
as the combination's secondary focal length /'. 

If we now apply the simple lens formula (Eq. 4a) to the rays as they 
enter and leave the second lens L 2 , we note that for the second lens 
alone f[ is the object distance (taken with a negative sign), /' is the image 
distance, and f 2 is the focal length. Applying Eq. 4a, these substitutions 
for s, s', and /, respectively, give 



J_ + i = I 

-/if /; 



or 



^ = J: + ^ 

r fi /; 



Having assumed the lenses are in air, the primary focal lengths are all 
equal to their respective secondary focal lengths, and we can drop all 
primes and write 

7 " T + T (4/l) 



56 



GEOMETRICAL OPTICS 



In words, the reciprocal of the focal length of a thin-lens combination is 
equal to the sum of the reciprocals of the focal le7igths of the individual lenses. 
Since by Eq. 4/ we can write P x = l/f h P 2 = V/2, and P = l/f, we 
obtain for the power of the combination 

p = Px + p* m 

In general, when thin lenses are placed in contact, the power of the combi- 
nation is given by the sum of the powers of the individual lenses. 



Q 








T 






\£ 








9 


^ 








(i 


W 


M 

l 




1 

1 




A 


.'/' 








Axis 




y 


1 


—x — 


1 
-*r* — 


— /- 


s 












^^ ' ' 


1 












( 


r 








' 









Fig. 4:K. The geometry used for the derivation of thin-lens formulas. 

4.14. Derivation of the Lens Formula. A derivation of Eq. 4a, the 
so-called "lens formula," is readily obtained from the geometry of Fig. 
4D. The necessary features bf the diagram are repeated in Fig. AK, 
which shows only two rays leading from the object of height y to the 
image of height y'. Let s and s' represent the object and image distances 
from the lens center and x and x' their respective distances from the focal 
points F and F' . 

From similar triangles Q'TS and F'TA the proportionality between 
corresponding sides gives 

y - y' _ y 

s' r 

Note that y - y' is written instead of y + y' because y', by the conven- 
tion of signs, is a negative quantity. From the similar triangles QTS 
and FAS, 

y - y' _ z^fL 
s / 

The sum of these two equations is 

y - y' , y - y' __ V tf_ 

~v~^ s' ~ r f 

Since / = /', the two terms on the right may be combined and y — y' 



THIN LENSES 57 

canceled out, yielding the desired equation, 

1 + 1 = 1 

s^ s' f 

This is the lens formula in the Gaussian* form. 

Another form of the lens formula, the so-called Newtonian form, is 
obtained in an analogous way from two other sets of similar triangles, 
QMF and FAS on the one hand, and TAF' and F'M'Q' on the other. 
We find 

Multiplication of one equation by the other gives 

xx' = P 

In the Gaussian formula the object distances are measured from the 
lens, while in the Newtonian formula they are measured from the focal 
points. Object distances (s or x) are positive if the object lies to the 
left of its reference point (4 or F, respectively), while image distances 
(s' or re') are positive if the image lies to the right of its reference point 
(A or F', respectively). 

The lateral magnification as given by Eq. 4c corresponds to the 
Gaussian form. When distances are measured from focal points, one 
should use the Newtonian form, which may be obtained directly from 
Eq. 4j. 

m -t„-l.* (4k) 

y x f 

In the more general case where the medium on the two sides of the 
lens is different, it will be shown in the next section that the primary and 
secondary focal distances / and /' are different, being in the same ratio 
as the two refractive indices. The Newtonian lens formula then takes 
the symmetrical form 

xx' - //' 

4.16. Derivation of the Lens Makers' Formula. The geometry 
required for this derivation is shown in Fig. 4L. Let n, n', and n" 
represent the refractive indices of the three media as shown, f\ and /( the 
focal lengths for the first surface alone, and f 2 and /" the focal lengths 

* Karl Friedrich Gauss (1777-1855). German astronomer and physicist, chiefly 
known for his contributions in the mathematical theory of magnetism. Coming 
from a poor family, he received support for his education because of his obvious 
mathematical ability. In 1841 he published the first general treatment of the first- 
order theory of lenses in his now famous papers, " Dioptrische Untersuchungen." 



58 



GEOMETRICAL OPTICS 



for the second surface alone. The oblique ray MT\ is incident on the 
first surface as though it came from an axial object point M at a distance 
Si from the vertex A\. At 7\ the ray is refracted according to Eq. 36 
and is directed toward the conjugate point M'. 



— _j_ _ — n n 
Si s[ " r x 



(4Z) 



Arriving at Tz, the same ray is refracted in the new direction T-M" . 
For this second surface the object ray T X T% has for its object distance 




Fig. 4L. Each surface of a thin lens has its own focal points and focal lengths, as well 
as separate object and image distances. 

s 2 , and the refracted ray gives an image distance of s" . When Eq. 36 
is applied to this second refracting surface, 



s' 2 + s" r 2 



(4m) 



If we now assume the lens thickness to be negligibly small compared 
with the object and image distances, we note the image distance s[ for the 
first surface becomes equal in magnitude to the object distance s' 2 for 
the second surface. Since M' is a virtual object for the second surface, 
the sign of the object distance for this surface is negative. As a conse- 
quence we can set s[ = —s' 2 and write 



?( 



W 



If we now add Eqs. U and 4m and substitute this equality, we obtain 



n .n _ n — n n — n 



Sl 



ri 



r-i 



(4n) 



THIN LENSES 



59 



If we now call Si the object distance and designate it s as in Fig. 4M, 
and call s' 2 ' the image distance and designate it s", we can write Eq. 4n 
as follows: 



n , n_ _ n' — n n" — n' 
s + s" = n r 2 



(4o) 



This is the general formula for a thin lens having different media 
on the two sides. For such cases we can follow the procedure given in 
Sec. 3.4 and define primary and secondary focal points F and F", and 




Fig. AM. When the media on the two sides of a thin lens have different indices, the 
primary and secondary focal lengths are not equal and the ray through the lens center 
is deviated. 

the corresponding focal lengths / and /", by setting s or s" equal to 
infinity. When this is done, we obtain 



n n — n , n 


- n' 


n 


7= n +~ 


Ti 


f 



(4p) 



In words, the focal lengths have the ratio of the refractive indices 
of the two media n and n" (see Fig. 4M). 



L 
f" 



n 
n" 



(4g) 



If the medium on both sides of the lens is the same, n = n", Eq. 4o 
reduces to 






•s 



8 



(4r) 



Note: The minus sign in the last factor arises when n" and n' are 
reversed for the removal of like terms in the last factor of Eq. 4o. 

Finally, in case the surrounding medium is air (w = 1), we obtain the 
lens makers' formula 



i + ^^-Dfl-I) 

s s" K \n r 2 / 



(4s) 



60 GEOMETRICAL OPTICS 

In the power notation of Eq. Si, the general formula (Eq. 4o), can be 
written 

V + V" = Pi + P 2 (40 

where V = - V" = —rj Pi = Pi = (4u) 

s s ri r% 

Equation 4£ can be written 

V + V" = P (4») 

where P is the power of the lens and is equal to the sum of the powers 
of the two surfaces. 

P = Pi + P* (4u>) 

PROBLEMS 

1. An object, located 16 cm in front of a thin lens, has its image formed on the 
opposite side 48 cm from the lens. Calculate (a) the focal length of the lens, and 
(6) the lens power. 

2. An object 2 cm high is placed 10 cm in front of a thin lens of focal length 4 cm. 
Find (a) the image distance, (b) the magnification, and (c) the nature of the image. 
Solve graphically and by calculation. 

Ans. (a) +6.66 cm. (b) —0.66. (c) Real and inverted. 

3. The two sides of a thin lens have radii ri = +12 cm and r 2 = —30 cm, respec- 
tively. The lens is made of glass of index 1.600. Calculate (a) the focal length, and 
(b) the power of the lens. 

4. An object 4 cm high is located 20 cm in front of a lens whose focal length / = 
— 5 cm. Calculate (a) the power of the lens, (6) the image distance, and (c) the lateral 
magnification. Graphically locate the image by (d) the parallel-ray method, and (e) 
the oblique-ray method. Ans. (a) -20 D. (6) -4.0 cm. (c) +0.20. 

6. An equiconcave lens is to be made of crown glass of index n = 1.65. Calculate 
the radii of curvature if it is to have a power of —2.5 D. 

6. A plano-convex lens is to be made of flint glass of index n = 1.71. Calculate 
the radius of curvature necessary to give the lens a power of +5 D. Ans. +14.2 cm. 

7. Two lenses with focal lengths /i = +12 cm and/ 2 = +24 cm, respectively, are 
located 6 cm apart. If an object 2 cm high is located 20 cm in front of the first lens, 
find (a) the position of, and (b) the size of the final image. 

8. A converging lens is used to focus the image of a candle flame on a distant screen. 
A second lens with radii r x = +12 cm and r 2 = —24 cm and index n = 1.60 is placed 
in the converging beam 40 cm from the screen. Calculate (a) the power of the second 
lens, and (ft) the position of the final image. Ans. (a) +7.5 D. (6) +10.0 cm. 

9. A double-convex lens is to be made of glass having a refractive index of 1.52. If 
one surface is to have twice the radius of the other and the focal length is to be 5 cm, 
find the radii. 

10. Two lenses having focal lengths /i = +8 cm and / 2 = — 12 cm are placed 6 cm 
apart. If an object 3 cm high is located 24 cm in front of the first lens, find (a) the 
position, and (6) the size of the final image. Ans. (a) +12.0 cm. (6) —3.0 cm. 

11. A lantern slide 2 in. high is located 10.5 ft from a projection screen. What 
is the focal length of the lens that will be required to project an image 40 in. high? 



THIN LENSES 61 

12. An object is located 1.4 m from a white screen. A lens of what focal length will 
be required to form a real and inverted image on the screen with a magnification of —6? 

Ans. +17.1 cm. 

13. Three thin lenses have the following powers: +1.0 D, —2.0 D, and +4.0 D. 
What are all the possible powers obtainable with these lenses using one, two, or three 
at a time in contact? 

14. Two thin lenses having the following radii of curvature and index are placed in 
contact: For the first lens r t =■ +16 cm, r 2 = —24 cm, and n = 1.60, and for the 
second lens r\ = —32 cm, n = +48 cm, and n = 1.48. Find their combined (a) 
focal length, and (b) power. Ans. (a) +26.6 cm. (6) +3.75 D. 

15. An object 2 cm high is located 12 cm in front of a lens of +4 cm focal length, 
and a lens of —8 cm focal length is placed 2 cm behind the converging lens. Find (a) 
the position, and (b) the size of the final image. 

16. An object 2 cm high is located 6 cm in front of a lens of —2 cm focal length. 
A lens of +4 cm focal length is placed 4 cm behind the first lens. Find (a) the position, 
and (o) the size of the final image. Ans. (a) +14.7 cm. (6) —1.33 cm. 

17. Three lenses with focal lengths of +12 cm, — 12 cm, and +12 cm, respectively, 
are located one behind the other 2 cm apart. If parallel light is incident on the first 
lens, how far behind the third lens will the light be brought to a focus? 

18. An object 4 cm high is located 10 cm in front of a lens of +2 cm focal length. 
A lens of —3 cm focal length is placed 12.5 cm behind the first lens. Find (a) the 
position, and (6) the size of the final image. 

Ans. (a) —2.31 cm from second lens, (b) —0.23 cm. 



CHAPTER 5 

THICK LENSES 



When the thickness of a lens cannot be considered as small compared 
with its focal length, some of the thin-lens formulas of Chap. 4 are no 
longer applicable. The lens must be treated as a thick lens. This term 
is used not only for a single homogeneous lens with two spherical surfaces 
separated by an appreciable distance but also for any system of coaxial 
surfaces which is treated as a unit. The thick lens may therefore include 
several component lenses, which may or may not be in contact. We 
have already investigated one case which comes under this category, 
namely, the combination of a pair of thin lenses spaced some distance 
apart as was shown in Fig. 47/. 




Fig. 5 A. Details of the refraction of a ray at both surfaces of a lens. 

6.1. Two Spherical Surfaces. A simple form of thick lens comprises 
two spherical surfaces as shown in Fig. 5 A. A treatment of the image- 
forming capabilities of such a system follows directly from procedures 
outlined in Chaps. 3 and 4. Each surface, acting as an image-forming 
component, contributes to the final image formed by the system as a 
whole. 

Let n, n f , and n" represent the refractive indices of three media sep- 
arated by two spherical surfaces of radius ri and r 2 . A light ray from an 
axial object point M is shown refracted by the first surface in a direction 
TiM' and then further refracted by the second surface in a direction 
TzM" . Since the lens axis may be considered as a second ray of light 
originating at M and passing through the system, M" is the final image 
of the object point M. Hence M and M" are conjugate points for the 
thick lens as a whole, and all rays from M should come to a focus at M". 

We shall first consider the parallel-ray method for graphically locating 

62 



THICK LENSES 



63 



an image formed by a thick lens and then apply the general formulas 
already given for calculating image distances. The formulas to be used 
are (see Sec. 3.4) 



n n _ 



n — n 



for first surface 



s 2 + s' 8 ' r 2 

for second surface 



(5a) 



6.2. The Parallel-ray Method. The parallel-ray method of graphical 
construction, applied to a thick lens of two surfaces, is shown in Fig. bB. 



(a) M 




(b) M 



Fig. 5B. Illustrating the parallel-ray method for graphically locating the image formed 
by a thick lens. 

Although the diagram is usually drawn as one, it has been separated into 
two parts here to simplify its explanation. The points F x and F[ repre- 
sent the primary and secondary focal points of the first surface, and F' 2 
and F'l represent the primary and secondary focal points of the second 
surface, respectively. 

Diagram (a) is constructed by applying the method of Fig. ZF to the 
first surface alone and extending the refracted rays as far as is necessary 
to locate the image M'Q' . This real image, M'Q', then becomes the 
object for the second surface as shown in diagram (6). The procedure 
is similar to that given for two thin lenses in Fig. 4#. Ray 5 in diagram 
(6), refracted parallel to the axis by the first surface, is refracted as ray 7 
through the secondary focal point F'{ of the second surface. 

Rays 8 and 9 are obtained by drawing a line from Q' back through C* 
and then, through the intersection B, drawing the line BQ. The inter- 
section of rays 7 and 8 locates the final image point Q" and the final 
image M"Q". 



64 GEOMETRICAL OPTICS 

Example 1 : An equiconvex lens 2 cm thick and having radii of curvature 
of 2 cm is mounted in the end of a water tank. An object in air is 
placed on the axis of the lens 5 cm from its vertex. Find the position 
of the final image. Assume refractive indices of 1.00, 1.50, and 1.33 for 
air, glass, and water, respectively. 

Solution: The relative dimensions in this problem are approximately 
those shown in Fig. 5B(b). If we apply Eq. 5a to the first surface alone, 
we find the image distance to be, 

1.00 , 1.50 1.50 - 1.00 , . _. 

-= 1 — = = or Sj = +30 cm 

When the same equation is applied to the second surface, we note 
that the object distance is si minus the lens thickness, or 28 cm, and that 
since it pertains to a virtual object it has a negative sign. The substitu- 
tions to be made are, therefore, St = —28 cm, n' = 1.50, n" = 1.33, and 
r% = —2.0 cm. 

1.50 , 1.33 1.33 - 1.50 „ . nr 

-\ -ft = n or s 2 = +9-6 cm 



-28 ' s' 2 ' -2 

Particular attention should be paid to the signs of the various quantities 
in this second step. Because the second surface is concave toward the 
incident light, r 2 must have a negative sign. The incident rays in the 
glass belong to an object point M', which is virtual, and thus s' 2 , being 
to the right of the vertex A2, must also be negative. The final image is 
formed in the water (n" = 1.33) at a distance +9.6 cm from the second 
vertex. The positive sign of the resultant signifies that the image is 
real. 

It should be noted that Eqs. 5a hold for paraxial rays only. The 
diagrams in Fig. 5J5, showing all refraction as taking place at vertical 
lines through the vertices A y and A 2 , are likewise restricted to paraxial 
rays. 

5.3. Focal Points and Principal Points. Diagrams showing the char- 
acteristics of the two focal points of a thick lens are given in Fig. 5C. 
In the first diagram diverging rays from the primary focal point F emerge 
parallel to the axis, while in the second diagram parallel incident rays are 
brought to a focus at the secondary focal point F". In each case the 
incident and refracted rays have been extended to their point of inter- 
section between the surfaces. Transverse planes through these inter- 
sections constitute primary and secondary principal planes. These planes 
cross the axis at points H and H", called the principal points. It will 
be noticed that there is a point-for-point correspondence between the two 
principal planes, so that each is an erect image of the other, and of the 
same size. For this reason they have sometimes been called "unit 



THICK LENSES 



65 



planes." They are best defined by saying that the principal planes are 
two planes having unit positive lateral magnification. 

The focal lengths, as shown in the figure, are measured from the focal 
points F and F" to their respective principal points H and H" and not to 
their respective vertices Ai and A 2. If the medium is the same on both 



Primary 
'principal 
plane 



Secondary 
/principal 
I plane 




Fig. 5C. Ray diagrams showing the primary and secondary principal planes of a thick 
lens. 



HH" 



HH" 



HH 



HH 



HH 








Fig. 5D. Illustrating the variation of the positions of the primary and secondary 
principal planes as a thick lens of fixed focal length is subjected to "bending." 



sides of the lens, n" = n, the primary focal length / is exactly equal to the 
secondary focal length /". 

If the media on the two sides of the lens are different so that n" is not 
equal to n, the two focal lengths are different and have the ratio of their 
corresponding refractive indices. 



n_ 

n 



(1 
f 



(56) 



In general the focal points and principal points are not symmetrically 
located with respect to the lens but are at different distances from the 
vertices. This is true even when the media on both sides are the same 
and the focal lengths are equal. As a lens with a given material and 
focal length is "bent" (see Fig. 5D), departing in either direction from the 



66 



GEOMETRICAL OPTICS 



symmetrical shape of an equiconvex lens, the principal points are shifted. 
For meniscus lenses of considerable thickness and curvature, H and H" 
may be completely outside the lens. 

6.4. Conjugate Relations. In order to trace any ray through a thick 
lens, the positions of the focal points and principal points must first 
be determined. Once this has been done, either graphically or by compu- 
tation, the parallel-ray construction can be used to locate the image 




Fio. 52?. Illustrating the parallel-ray method of construction for graphically locating 
an image formed by a thick lens. 




Fig. 5F. Principal planes and antiprincipal planes are planes of unit magnification. 



as shown in Fig. 5E. The construction procedure follows that given in 
Fig. AM for a thin lens, except that here all rays in the region between 
the two principal planes are drawn parallel to the axis. 

By a comparison of the two figures, and the derivations of Eqs. 4w 
and 4o, it will be found that, provided the object and image distances 
are measured to or from the principal points, we may apply the Gaussian 
lens formula 



or by Eq. 3/i 



n , n_ _ n _ n 
s a" " / ~ /" 



V + 7" = P 



(6c) 



M 



THICK LENSES 



67 



In the special case where the media on the two sides of the lens are 
the same, so that n" = n, we find /" = / and Eq. (5c) becomes 



1 j _L I _L 



(5e) 



Figure 5F shows that for the purposes of graphical construction the 
lens may be regarded as replaced by its two principal planes. 

6.6. The Oblique-ray Method. The oblique-ray method of construc- 
tion may be used to find graphically the focal points of a thick lens. As 
an illustration, consider a glass lens of index 1.50, thickness 2 cm, and 



F, 




Fig. 5G. Illustrating the oblique-ray method for graphically tracing paraxial rays 
through a thick lens. 

radii ri = +3 cm, r 2 = — 5 cm, surrounded by air of index n = 1.00. 
The first step is to calculate the primary and secondary focal lengths 
of each surface separately by the use of the formulas for a single spherical 
surface (Eqs. 3c and 3d). Using the present notation, these are 






n 



n — n 
ri 



, n' n" n" - n' ,_. 

and — = — = — (5/j 



The given quantities are 

ri = +3 cm r 2 = — 5 cm d = 2 cm n' = 1.50 n" = n = 1.00 

By substituting these values in Eqs. 5/, we obtain 

/i = +6 cm f[ = +9 cm /' 2 = +15 cm / 2 ' = +10 cm 

With these focal lengths known, the lens axis can be drawn as in Fig. 5G 
and the known points measured off to some suitable scale. After drawing 
lines 2 and 3 through the lens vertices, a parallel incident ray 4 is selected. 
Upon refraction at the first surface the ray takes the new direction 5 
toward F[, the secondary focal point of that surface. After line 6 is 
drawn through F' 2 ', line 7 is drawn through C 2 parallel to ray 5. The 
point B where line 6 crosses line 7 determines the direction of the final 



68 GEOMETRICAL OPTICS 

refracted ray 8. The intersection of ray 8 with the axis locates the 
secondary focal point F" of the lens, while its intersection N" with the 
incident ray locates the corresponding secondary principal plane H" . 

By turning the lens around and repeating this procedure, the position 
of the primary focal point F and the position of the primary principal 
point H can be determined. The student will find it well worthwhile 
to carry out this construction and to check the results by measuring the 
focal lengths to verify the fact that they are equal. It is to be noted 
that, in accordance with the assumption of paraxial rays, all refraction 
is assumed to occur at the plane tangent to the boundary at its vertex. 

6.6. General Thick-lens Formulas. There is a set of formulas that 
may be used for the calculation of important constants generally associ- 
ated with a thick lens. These formulas are presented below in the form 
of two equivalent sets. 



Gaussian Formulas 




Power Formulas 




n n' n" dn" 


n" 

r 


P - Pi + P 2 - ~, PxP* 

n 


(50 


*r--*(i-jD 




A, — i(l-$p) 


(5h) 


A X E = +ff, 




**-+fy p ' 


M 


A*F" = +/" (l - |) 




«"-+t( 1 -^) 


m 


AJ1" = -/" i 
Ji 




A 2 H" = -!£.*P l 
P n 


(5k) 



These equations are derived from geometrical relations that may be 
obtained from a diagram like Fig. 5G. As an illustration, the Gaussian 
equation 5/c is derived as follows: From the two similar right triangles 
TiAiF[ and TiAiF' x , we can write corresponding sides as proportions 

ArF[ Atf[ nr ft _ f[-d 
AxTx A 2 T 2 h j 

and, from the two similar right triangles N"H"F" and T2A2F", we can 
write the proportions 

H"F" A2F" f" f" - H"A, 

or -T- - 



H"N" A 2 T 2 h j 

If we solve each equation for j/h and then equate the right-hand sides 
of the resultant equations, we obtain 



THICK LENSES 69 

If we now reverse the segment H"A 2 to A 2 H" by changing the sign from 
+ to — , we obtain 

AJ1" = -/" j, 

J\ 

In terms of surface power and lens power, 

„ n n' n' n" D n n" ,_ n 

the same equation can be written 

In the design of certain optical systems it is convenient to know the 
vertex power of a lens. This power, sometimes called effective power, is 
given as 

p - = i - &v»> (5m) 

and is defined as the reciprocal of the distance from the back surface 
of the lens to the secondary focal point. This distance is commonly 
called the back focal length. Since P v = I/A2F", the above equation for 
vertex power is obtained by inverting Eq. 5j. In the inversion the lens 
is assumed to be in air so that n" = 1. 

In a similar way the distance from the primary focal point to the 
front surface of the lens is called the front focal length, and the reciprocal 
of this distance is called the neutralizing power. P n = 1/AiF. Calling 
P n the neutralizing power, we can take the reciprocal of Eq. 5h to obtain 

p - - 1 - £w (5re) 

The name is derived from the fact that a thin lens of this specified 
power and of opposite sign will, upon contact with the front surface, 
give zero power to the combination. 

The following example will serve as an illustration of the use of thick- 
lens formulas applied to two surfaces: 

Example 2: A lens has the following specifications: r t = +1.5 cm, 
r 2 = +1.5 cm, d = 2.0 cm, n = 1.00, n' = 1.60, and n" = 1.30. Find 
(a) the primary and secondary focal lengths of the separate surfaces, 
(6) the primary and secondary focal lengths of the system, and (c) the 
primary and secondary principal points. 



70 GEOMETRICAL OPTICS 

Solution: To apply the Gaussian formulas, we first calculate the indi- 
vidual focal lengths of the surfaces by means of Eq. 5/. 

n n' - n 1.60 - 1.00 , 1.00 , _ „ 

7, " —FT = lJ— h = 616 = +25 ° cm 

= 0.400 ,, 1.60 . . ftn 

* " 0M ' +4 -°° Cm l Ans. 
n' n" - n' 1.30 - 1.60 ,, 1.60 / (a) 

= -0.200 .„ 1.30 

h = ^O20 = - 650cm 

The focal lengths of the system are calculated from Eq. hg. 

n n' n" d n" 1.60 1.30 2.00 1.30 



/ /( /a' /i/" 4.00 ' -6.50 4.00 -6.50 
j « 0.40 - 0.20 + 0.10 = 0.30 
or 

' - sS - + 3 - 333 cm and '" - rao - m = + 4 - 333 cm 

The focal points of the system are given by Eqs. 5h and 5j. 

AiF " "K 1 " S " ~ 3 - 333 ( ! _ ^o) = ~ 4166 

Ai F" = +/" f 1 - jrf - +4.33 (l - ||j - +2.167 



cm. 



^4ns. (6) 



cm 



The principal points are given by Eqs. 5i and 5fc. 

Axil = +/| = +3.33 -^r = -0.833 cm) 

d _ 2^ ^ « 

A 2 H" = -j"^j = -4.33 ^ = -2.167 cm \ 

Positive signs represent distances measured to the right of the reference 
vertex and negative signs those measured to the left. By subtracting 
the magnitudes of the two intervals A\F and A\H, the primary focal 
length FH = 4.166 — 0.833 = 3.333 cm is obtained and serves as a check 
on the calculations in part (6). Similarly the addition of the two intervals 
A2F" and A2H" gives the secondary focal length 

H"F" = 2.167 + 2.167 = 4.334 cm 

The graphical solution of this same problem is shown in Fig. 5H. 
After the axis is drawn and the lens vertices Ai and A% and the centers 



THICK LENSES 



71 



Ci and C 2 are located, the individual focal points F h F[, F' 2t and F'l are 
laid off according to the results in part (a). The parallel ray (1) is 
refracted at the first surface toward F[. The oblique-ray method is 
applied to this ray (2) at the second surface, and the final ray (3) is 
obtained. The point where ray (3) crosses the axis locates the secondary 
focal point F", and the point where its backward extension intersects 




Fig. 5H. A graphical construction for locating the focal points and principal points 
of a thick lens. 




Fig. 57. Special thick lenses, (a) A positive lens with equal radii of curvature, {b) 
A negative lens with concentric surfaces. 

ray (1) locates the secondary principal plane H". Ray (4) is constructed 
backward by drawing it parallel to the axis and from right to left. The 
first refraction gives ray (5) up and to the left as if it came from F' 2 . The 
oblique-ray method applied to ray (5) at the left-hand surface yields 
ray (6) . The point where ray (6) crosses the axis locates F, and the point 
where it crosses the extension of ray (4) locates H. Hence parts (b) and 
(c) of the problem are solved graphically, and they check with the 
calculated values. 

5.7. Special Thick Lenses. Two special lenses of some interest as 
well as practical importance are presented here. The first, as shown 
in Fig. 57, is a lens with spherical surfaces of equal radii, r t = r 2 . A lens 



72 



GEOMETRICAL OPTICS 



of this description, surrounded by a medium of lower index, n' > n, has 
a small but positive power. Its principal planes are located some dis- 
tance from and to the right of the lens, and their spacing HH" is equal 
to the lens thickness d. If the surrounding medium has a greater index, 
as in the case of an air space between the surfaces of two lenses of equal 
index, n' < n, the power is again positive but the principal planes lie 
some distance to the left of the lens and a distance d apart. 

The second special case is that of a concentric lens, both surfaces 
having the same center of curvature. Where such a lens is surrounded 
by a medium of lower index, n' > n, the system has a negative power 
with a long focal length and the principal points coincide with the common 
center of curvature of the two surfaces. In other words, it acts like a 
thin lens located at C1C2. 

5.8. Nodal Points and Optical Center. Of all the rays that pass 
through a lens from an off-axis object point to its corresponding image 

point, there will always be one for 
which the direction of the ray in the 
image space is the same as that in 
the object space, i.e., the segments 
of the ray before reaching the lens, 
and after leaving it, are parallel. The 
two points at which these segments, 
if projected, intersect the axis are 
called the nodal points, and the trans- 
verse planes through them are called 
the nodal planes. This third pair : of 
points and their associated planes are 
shown in Fig. 5J, which also shows the optical center of the lens at C. It is 
readily shown that if the medium on both sides is the same, the nodal points 
N and N" coincide with the principal points H and H" , but that if the two 
media have different indices, the principal points and the nodal points will 
be separate. Since the incident and emergent rays make equal angles with 
the axis, the nodal points are called conjugate points of unit positive angular 
magnification. 

If the ray is to emerge parallel to its original direction, the two surface 
elements of the lens where it enters and leaves must be parallel to each 
other so that the effect is like that of a plane-parallel plate. A line 
between these two points crosses the axis at the optical center C. It is 
therefore through the optical center that the undeviated ray must be 
drawn in all cases. It has the interesting property that its position, 
depending as it does only on the radii of curvature and thickness of the 
lens, does not vary with color of the light. All the six cardinal points 
(Sec. 5.9) will in general have a slightly different position for each color. 




Fig. 5/. Illustrating the significance 
of the nodal points and nodal planes 
of a thick lens. 



THICK LENSES 



Figure 5K will help to clarify the different significance of the nodal 
points and the principal points. This figure is drawn for n" 9^ n, so 
that the two sets of points are separate. Ray 1 1 through the secondary 
nodal point is parallel to ray 10, the latter being incident in the direction 
of the primary nodal point iV. On the other hand both these segments 
intersect the principal planes at the same distance above the principal 
points H and H" . From the small parallelogram at the center of the 




Fig. 5K. Illustrating the parallel-ray method of graphically locating the nodal points 
and planes of a thick lens. 

diagram, it is observed that the distance between nodal planes is exactly 
equal to the distance between principal planes. In general, therefore, 

NN" = HH" (00) 

Furthermore in this case, where the initial and final values of the refrac- 
tive index differ, the focal lengths, which are measured from the principal 
points, are no longer equal. The primary focal length FH is equal to 
the distance N"F", while the secondary focal length H"F" is equal 
to FN: 

f = FH = N"F" and /" = H"F" = FN (5p) 

Nodal points may be determined graphically, as shown in Fig. 5K, by 
measuring off the distance ZQ = HH" = Z'Q" and drawing straight 
lines through QZ' and ZQ". From the geometry of this diagram, the 
lateral magnification y'/y is given by 



m = - 



•J 



a" - HN 
s + HN 



where 



HN =/' 



n" - n 



(oq) 
(or) 



When the object and image distances s and s" are, as usual, measured 
from their corresponding principal points H and H", Eq. 5c is valid for 
paraxial rays. 



74 GEOMETRICAL OPTICS 

The distance from the first vertex to the primary nodal point is given by 



^='d + ^p- B ) 



(5s) 



Example 3: Find the nodal points of the thick lens given in Example 2. 

Solution: To locate the primary nodal point N, we may use Eq. 5r and 
substitute the given values of n = 1.00 and n" = 1.30 and the already 
calculated value of /" = +4.333 cm, 

HN = 4.333 ( 1 - 30 1 - 100 ) = +1.00 cm 

Hence the nodal points N and N" are 1.00 cm to the right of their 
respective principal points H and H". 

5.9. Other Cardinal Points. In thick-lens problems a knowledge of 
the six cardinal points, comprising the focal points, principal points, and 
nodal points, is always adequate to obtain solutions. Other points of 
lesser importance but still of some interest are (1) negative principal 
points, and (2) negative nodal points. Negative principal points are con- 
jugate points for which the lateral magnification is unity and negative. 
For a lens in air they lie at twice the focal length and on opposite sides 
of the lens. Negative nodal points lie as far from the focal points as the 
ordinary cardinal nodal points, but on opposite sides. Their position is 
such that the angular magnification is unity and negative. Although a 
knowledge of these two pairs of cardinal points is not essential to the 
solution of optical problems, in certain cases considerable simplification 
is achieved by using them. 

5.10. Thin-lens Combination as a Thick Lens. A combination of two 
or more thin lenses may also be referred to as a thick lens. This is 
because of the fact that the optical properties of a set of coaxially mounted 
lenses can be conveniently treated in terms of only two focal points and 
two principal points. If the object space and image space have the same 
refractive index (and this is nearly always the case), the nodal points and 
planes coincide with the principal points and planes. 

A combination of two thin lenses with focal lengths of 8 and 9 cm 
respectively is shown in Fig. 5L. By the oblique-ray method the focal 
points F and F" and the principal points H and H" have been determined 
graphically. In doing so the refraction at each lens was considered in 
the same way as the refraction at the individual surfaces of the thick 
lens of Fig. 5G. There is a strong resemblance between these two dia- 
grams; i.e., for a thin lens we assume that all of the deviation occurs 
at one plane, just as for a single surface. This assumption is justified 
only when the separation of the principal planes of the lens can be neg- 



THICK LENSES 



75 



lected. The definition of a thin lens is just a statement of this fact: 
a thin lens in one in which the two principal planes and the optical center 
coincide at the geometrical center of the lens. The locations of the centers 
of the two lenses in this example are labeled Ai and A 2 in Fig. 5L. 

A diagram for a combination of a positive and a negative lens is given 
in Fig. 5M. The construction lines are not shown, but the graphical pro- 
cedure used in determining the paths of the two rays is the same as that 




\l 6 

L x L 2 

Fig. 5L. Focal points and principal points of a system involving two thin lenses. 

H H" 




Fig. 5Af . Illustrating the oblique-ray method as applied to positive and negative thin 
lenses in combination. 

shown in Fig. 5L. Note here that the final principal points H and H" 
he outside the interlens space but that the focal lengths / and /" measured 
from these points are as usual equal. The lower ray, although shown 
traveling from left to right, is graphically constructed by drawing it from 
right to left. 

The positions of the cardinal points of a combination of two thin lenses 
in air can be calculated by means of the thick-lens formulas given in Sec. 
5.6. As used for thin lenses in place of individual refracting surfaces, 
A i and A 2 become the two lens centers, while /i, / 2 and Pi, P 2 become 
their separate focal lengths and powers, respectively. The latter are 
given by 



76 



GEOMETRICAL OPTICS 



„ ni — n n' — ni n D n 2 — n' n" — n 2 n' ,„,. 

where r x and r[ are the radii of the first lens of index n h and r 2 and K 
are the radii of the second lens of index n 2 . The surrounding media have 
indices n, n', and n" (see Fig. 5L). The other formulas, Eqs. 5^, 5h, 
5i, 5j, and 5k, remain unchanged. 

To illustrate the use of these formulas, let us consider the following 
problem on a lens combination similar to that shown in Fig. 5M: 

Example 4: An equiconvex lens with radii of 4 cm and index n x = 1.50 
is located 2 cm in front of an equiconcave lens with radii of 6 cm and 
index n 2 = 1.60. The lenses are to be considered as thin. The sur- 
rounding media have indices n = 1.00, n' = 1.33, andw" = 1.00. Find 
(a) the power, (b) the focal lengths, (c) the focal points, and (d) the 
principal points of the system. 

Solution: In this instance we shall solve the problem by the use of 
the power formulas. By Eqs. 51, the powers of the two lenses in their 
surrounding media are 



0.04 



-0.04 



P 2 = 



1.60 - 1.33 , 1.00 - 1.60 



^ + 



0.0 



= -4.45 - 10.0 = - 14.45 D 



-0.06 

By Eq. 5g, we obtain 

P = 16.67 - 14.45 + 0.015 X 16.67 X 14.45 
P = +5.84 D Ans. (a) 



or 



Using Eq. 51, we find 



f" = 



1.00 
3.84 
1.00 



n 
f = -p = T~ni — 0.171 m = 17.1 cm 



= 0.171 m = 17.1 cm 



Ans. (6) 



P 5.84 
By Eqs. 5h, 5i, 5j, and 5k, we obtain 

AlF = ~ ZM (1 + °- 015 X 14 ' 45) = -°- 208 m = ~ 20 - 8 cm 



AiH = + 1:^0.015 (-14.45) 
1.00 



= -0.037 m = -3.7 cm 



AtF" = + ^g4 C 1 - 0.015 X 16.67) = +0.128 m = +12.8 cm 



AtH" = - ^°r 0.015 X 16.67 

O.o4 



Ans. (c) 



= -0.043 m = -4.3 cm 



Ans. (d) 



THICK LENSES 



77 



As a check on these results we find that the difference between the first 
two intervals A X F and AJi gives the primary focal length FH = 17.1 cm. 
Similarly the sum of the second two intervals A^F" and AzH" gives the 
secondary focal length H"F" = 17.1 cm. 

5.11. Thick-lens Combinations. The problem of calculating the posi- 
tions of the cardinal points of a thick lens consisting of a combination of 
several component lenses of appreciable thickness is one of considerable 
difficulty, but one which may be solved by use of the principles already 
given. If, in a combination of two lenses such as that in Fig. 5L, the 
individual lenses cannot be considered as thin, each must be represented 



M 




Fig. 5A". Illustrating the use of the nodal slide in locating nodal points. 



by a pair of principal planes. There are thus two pairs of principal 
points, Hi. and H[ for the first lens and H' 2 and H' 2 ' for the second, and 
the problem is to combine these to find a single pair H and H" for the 
combination, and to determine the focal lengths. By carrying out a 
construction similar to Fig. 5G for each lens separately, it is possible to 
locate the principal points and focal points of each. Then the construc- 
tion of Fig. 5L may be accomplished, taking account of the unit magnifi- 
cation between principal planes. 

Formulas may be given for the analytical solution of this problem, but 
because of their complexity they will not be given here.* Instead, we 
shall describe a method of determining the positions of the cardinal points 
of any thick lens by direct experiment. 

5.12. Nodal Slide. The nodal points of a single lens, or of a com- 
bination of lenses, may be located experimentally by mounting the system 
on a nodal slide. This is merely a horizontal support which permits rota- 
tion of the lens about any desired point on its axis. As is shown in Fig. 
5N, light from a source S is sent through a slit Q, adjusted to He at the 
secondary focal point of the lens. Emerging as a parallel beam, this 

* These equations are given, for example, in G. S. Monk, "Light, Principles and 
Experiments," 1st ed., McGraw-Hill Book Company, Inc., New York, 1937. 



78 



GEOMETRICAL OPTICS 



light is reflected back on itself by a fixed plane mirror M, passing again 
through the lens system and being brought to a focus at Q". This image 
of the slit is formed slightly to one side of the slit itself on the white face 
of one of the slit jaws. The nodal slide carrying the lens system is now 




Fig. 50. Rotation of a lens about its secondary nodal point shifts the refracted rays 
but not the image. 

rotated back and forth and the lens repeatedly shifted, until the rotation 
produces no motion of the image Q" . When this condition is reached, 
the axis of rotation N" locates one nodal point. By turning the nodal 
slide end-for-end and repeating the process, the other nodal point N is 

found. When performed in air, this 
experiment of course locates the prin- 
cipal points as well, and the distance 
N"Q" is an accurate measure of the 
focal length. 

The principle of this method of ro- 
tation about a nodal point is illus- 
trated in Fig. 50. In the first diagram 
ray 4 along the axis passes through 
AT and N" to the focus at Q". In the 
second diagram the lens system has 
been rotated about N" and the same 
bundle of rays passes through the 
lens, coming to a focus at the same 
point Q". Ray 3 is now directed 
toward N and ray 4 toward N". 
When projected across from the plane 
of N to that of N", the rays still converge toward Q" even though F" is 
now shifted to one side. Note that ray 3 approaches N in exactly the 
same direction that it leaves N", corresponding to the defining condition 
for the nodal points. 

If a camera lens is pivoted about its secondary nodal point and a long 
strip of photographic film is curved to a circular arc of radius /", a con- 
tinuous picture covering a very wide angle may be taken. Such an 
instrument, shown schematically in Fig. 5P, is known as a panoramic 




Fig. 5P. In the panoramic camera the 
lens rotates about a nodal point as a 
center. 






THICK LENSES 79 

camera. The shutter usually consists of a vertical slit just in front of 
the film, which moves with the rotation so that it always remains cen- 
tered on the lens axis. 



PROBLEMS 

1. An equiconvex lens has an index of 1.80, radii of 4.0 cm, and a thickness of 
3.6 cm. Calculate (a) the focal length, (6) the power, and (c) the distances from the 
vertices to the corresponding focal points and principal points. 

2. Solve Prob. 1 graphically, locating the focal points and principal points. 

Ans. A X F = -1.87 cm. AiH = +1.25 cm. AiF" = +1.87 cm. AiH" - 
-1.25 cm. 

3. A plano-convex lens 3.2 cm thick is made of glass of index 1.60. If the second 
surface has a radius of 3.2 cm, find (a) the focal length of the lens, (b) the power, and 
(c) the distances from the vertices to the corresponding focal points and principal 
points. 

4. Solve Prob. 3 graphically, locating the focal points and principal points. 

Ans. AiF = -3.33 cm. A X H = +2.00 cm. AiF" = +5.33 cm. A 2 H" = 0. 

6. A glass lens with radii r\ =* +3.0 cm and r 2 = +5.0 cm has a thickness of 

3.0 cm and an index of 1.50. Calculate (a) the focal length, (6) the power, and (c) 

the distances from the vertices to the corresponding focal points and principal points. 

6. Solve Prob. 5 graphically, locating the focal points and principal points. 

Ans. AiF = -12.0 cm. AiH = -2.0 cm. AiF" = +6.66 cm. AiH" = -3.33 
cm. 

7. A glass lens with radii r\ = +5.0 cm and r 2 = +2.5 cm has a thickness of 3.0 
cm and an index of 1.50. Calculate (a) the focal length, (6) the power, and (c) the 
distances from the vertices to the corresponding focal points and principal points. 

8. Solve Prob. 7 graphically, locating the focal points and principal points. Use 
the method outlined in Fig. 5H. 

Ans. AiF = +23.33 cm. A X H = +6.66 cm. AiF" = -13.33 cm. A 2 H" = 
+3.33 cm. 

9. A thick lens with radii n = —8.0 cm and r s = —4.0 cm has a thickness of 
3.23 cm and an index of 1.615. Calculate (a) the focal length, (6) the power, and 
(c) the distances from the vertices to the corresponding focal points and principal 
points. 

10. Solve Prob. 9 graphically, locating the focal points and principal points. Use 
the method shown in Fig. 5//. 

Ans. AiF = -6.93 cm. AiH = +3.07 cm. AiF" = +11.54 cm. A 2 H" = 
+ 1.54 cm. 

11. A thick glass lens is placed in the end of a tank containing an oil of refractive 
index 1.30. The lens, with radii ri = +4.2 cm and r 2 — —2.0 cm, is 5.1 cm thick 
and has a refractive index of 1.70. If r 2 is in contact with the oil, find (a) the primary 
and secondary focal length, (6) the power, and (c) the distances from the vertices to 
the corresponding focal points and principal points. 

12. Solve Prob. 11 graphically, locating the focal points and principal points. Use 
the method shown in Fig. 5H. 

Ans. AiF - -1.50 cm. AiH = +2.25 cm. AiF" = +2.44 cm. A 2 H" = -2.44 
cm. 

13. A glass lens 3.0 cm thick has radii t v = +5.0 cm and r 2 = +2.0 cm and an 
; ndex of 1.50. If r 2 is in contact with a liquid of index 1.40, find (a) the primary and 



80 GEOMETRICAL OPTICS 

secondary focal lengths, (6) the power, and (c) the distances from the vertices to the 
corresponding focal points, principal points, and nodal points. 

14. Solve Prob. 13 graphically, locating the six cardinal points of the lens. Use the 
methods of Figs. bll and 5A\ 

Ans. AiF = -18.33 cm. A X H = -1.66 cm. A t N = +5.0 cm. AiF" = 
+ 18.66 cm. A t H" - -4.66 cm. A t N" = +2.0 cm. 

16. A glass lens with radii ri = +4.0 cm and r 2 = +4.0 cm has an index of 1.50 
and a thickness of 1.5 cm. Oil of index 1.30 is in contact with r 2 . Find (a) the pri- 
mary and secondary focal lengths, (b) the power, and (c) the distances from the ver- 
tices to the corresponding focal points, principal points, and nodal points. 

16. Solve Prob. 15 graphically, locating the six cardinal points of the lens. Use the 
methods of Figs. bH and bK. 

Ans. AxF m -12.92 cm. A X H = -0.61 cm. AiN = +3.08 cm. AiF" = 
+ 14.00 cm. A t H" = -2.00 cm. AiN" = +1.69 cm. 

17. A glass lens 4.8 cm thick and index 1.60 has radii r\ = +6.0 cm and r 2 = +5.0 
cm. If a liquid of index 1.20 is in contact with ri and another liquid of index 2.0 is in 
contact with r 2 , find (a) the primary and secondary focal lengths, (6) the power, and (c) 
the distances from the vertices to the corresponding focal points, principal points, and 
nodal points. 

18. Solve Prob. 17 graphically, locating the six cardinal points of the lens. Use the 
method outlined in Figs. bH and bK. 

Am. AiF = -6.98 cm. A X H = +2.20 cm. A t N = +8.32 cm. AiF" = 
+ 12.24 cm. A 2 H" = -3.06 cm. A,N" = +3.06 cm. 

19. Two thin lenses, each with a focal length of +10.0 cm, are placed 4.0 cm apart 
in air. Find, for the combination, (a) the focal length, (6) the power, and (c) the 
distances from the lens centers to the focal points and principal points. 

20. Solve Prob. 19 graphically, locating the focal points and principal points. Use 
the method of Fig. bL. 

Ans. A X F = -3.75 cm. A\H = +2.50 cm. AiF" = +3.75 cm. A 2 H" = 
-2.50 cm. 

21. Two thin lenses with focal lengths /i = +10.0 cm and / 2 = —10.0 cm are 
located 5.0 cm apart in air. Find, for the combination, (a) the focal length, (6) the 
power, and (c) the distances from the lens centers to the focal points and principal 
points. 

22. Solve Prob. 21 graphically, locating the focal points and principal points. Use 
the method of Fig. bM . 

Ans. A X F = -30.0 cm. A t H = -10.0 cm. AtF" = +10.0 cm. A t H" = 
-10.0 cm. 

23. Two thin lenses with focal lengths /i = —10.0 cm and / 2 = +10.0 cm are 
located 5.0 cm apart in air. Find, for the combination, (a) the focal length, (6) the 
power, and (c) the distances from the lens centers to the focal points and principal 
points. 

24. Solve Prob. 23 graphically, locating the focal points and principal points. Use 
the method of Fig. bM. 

Ans. AiF = -10.0 cm. AiH = +10.0 cm. AJF" = +30.0 cm. AtH" = 
+ 10.0 cm. 

25. Two thin lenses with focal lengths of /i = —10.0 cm and /j = —20.0 cm are 
located 5.0 cm apart in air. Find, for the combination, (a) the focal length, (6) the 
power, and (c) the distances from the lens centers to the focal points and principal 
points. 

26. Solve Prob. 25 graphically, locating the focal points and principal points. Use 
the method of Fig. bL. 



THICK LENSES 81 

Ans. AiF = +7.14 cm. AxH = +1.43 cm. AJF" = -8.57 cm. A 2 H" = 
-2.86 cm. 

27. With Fig. bG as a guide make a diagram locating the secondary focal point. 
From similar triangles in your diagram derive Eq. 5j. 

28. With Fig. 5G as a guide make a diagram locating the primary focal point. 
From similar triangles in your diagram derive Eq. 5h. 

29. A lens with equal radii of curvature, r\ = r 2 = +5.0 cm, is 3 cm thick and has 
an index of 1.50. If the lens is surrounded by air, find (a) the power, (6) the focal 
length, and (c) the focal points and principal points. 

30. A concentric lens has radii of —5 cm and —8 cm, respectively, and an index of 
1.50. If the lens is surrounded by air, find (a) the power, (6) the focal length, and 
(c) the focal points and principal points. • 

Ans. (o) -2.5 D. (6) -40 cm. (c) AiF = +35.0 cm, AiH = -5.0 cm, A J"' = 
-48.0 cm, A Z H" = -8.0 cm. 



CHAPTER 6 

SPHERICAL MIRRORS 



A spherical reflecting surface has image-forming properties similar to 
those of a thin lens or of a single refracting surface. The image from a 
spherical mirror is in some respects superior to that from a lens, notably 
in the absence of chromatic effects due to dispersion that always accom- 
pany the refraction of white light. Therefore mirrors are occasionally 
used in place of lenses in optical instruments, but their applications are 
not so broad as those of lenses because they do not offer the same possi- 
bilities for correction of the other aberrations of the image (Chap. 9). 




Fig. 6 A. The primary and secondary focal points of spherical mirrors coincide. 

Because of the simplicity of the law of reflection as compared to the 
law of refraction, the quantitative study of image formation by mirrors 
is easier than in the case of lenses. Many features are the same, and 
these we shall pass over rapidly, putting the chief emphasis upon those 
characteristics which are different. To begin with, we restrict the discus- 
sion to images formed by paraxial rays. 

6.1. Focal Point and Focal Length. Diagrams showing the reflection 
of a parallel beam of light by a concave mirror and by a convex one are 
given in Fig. QA. A ray striking the mirror at some point such as T 
obeys the law of reflection <f>" = <f>. All rays are shown as brought to 
a common focus at F, although this will be strictly true only for paraxial 
rays. The point F is called the focal point and the distance FA the focal 
length. In the second diagram the reflected rays diverge as though they 

82 




SPHERICAL MIRRORS 83 

came from a common point F. Since the angle TCA also equals <£, the 
triangle TCF is an isosceles one, and in general CF = FT. But for very 
small angles 4> (paraxial rays), FT approaches equality with FA. Hence 

(FA) = UCA) or / = -*r (6a) 

and the focal length equals one-half the radius of curvature (see also 
Eq. 6d). 

The negative sign is introduced in Eq. 6a so that the focal length of a 
concave mirror, which behaves like a positive or converging lens, will also 
be positive. According to the sign convention of Sec. 3.5, the radius of 
curvature is negative in this case. The focal length of a convex mirror, 
which has a positive radius, will then 
come out to be negative. This sign 
convention is chosen as being con- 
sistent with that used for lenses; it 
gives converging properties to a mir- 
ror with positive / and diverging 
properties to a mirror with negative 
/. By the principle of reversibility 
it may be seen from Fig. 6 A that the 

primary and secondary focal points Fl °- 6B - Parallel rays ***** on a con - 
c . . _ , , cave mirror but inclined to the axis are 

of a mirror coincide. In Other words, brought to a focus in the focal plane, 
it has but one focal point. 

As before, a transverse plane through the focal point is called the focal 
plane. Its properties, as shown in Fig. 65, are similar to those of either 
focal plane of a lens ; for example, parallel rays incident at any angle with 
the optic axis are brought to a focus at some point in the focal plane. 
The image Q' of a distant off-axis point object occurs at the intersection 
with the focal plane of that ray which goes through the center of cur- 
vature C. 

6.2. Graphical Constructions. Figure 6C, which illustrates the forma- 
tion of a real image by a concave mirror, is self-explanatory. When the 
object MQ is moved toward the center of curvature C, the image also 
approaches C and increases in size until when it reaches C it is the same 
size as the object. The conditions when the object is between C and F 
may be deduced from the interchangeability of object and image as 
applied to this diagram. When the object is inside the focal point, the 
image is virtual as in the case of a converging lens. The methods of 
graphically constructing the image follow the same principles as were 
used for lenses, including the fact that paraxial rays must be represented 
as deflected at the tangent plane instead of at the actual surface. 

An interesting experiment can be performed with a large concave mirror 
set up under the condition of unit magnification, as shown in Fig. 6D. A 



84 



GEOMETRICAL OPTICS 



bouquet of flowers is suspended upside down in a box and illuminated by 
a shaded lamp S. The large mirror is placed with its center of curvature 
C at the top surface of the stand, on which a real vase is placed. The 
observer's eye at E sees a perfect reproduction of the bouquet, not merely 
as a picture but as a faithful three-dimensional replica, which creates a 
strong illusion that it is a real object. As shown in the diagram, the 




Fig. 6C. Real image due to a concave mirror. 
Real image _ _ 




Real flowers 

Fig. 6D. Experimental arrangement for an optical illusion produced by a real image 
of unit magnification. 

rays diverge from points on the image just as they would were the real 
object in the same position. 

The parallel-ray method of construction is given for the case of a con- 
cave mirror in Fig. QE. Three rays leaving Q are, after reflection, 
brought to the conjugate point Q' . The image is real, inverted, and 
smaller than the object. Ray 4 drawn parallel to the axis is, by defini- 
tion of the focal point, reflected through F. Ray 6 drawn through F is 
reflected parallel to the axis, and ray 8 through the center of curvature 
strikes the mirror normally and is reflected back on itself. The crossing 
point of any two of these rays is sufficient to locate the image. 



SPHERICAL MIRRORS 



85 



A similar procedure is applied to a convex mirror in Fig. 6F. The rays 
from the object point Q, after reflection, diverge from the conjugate point 
Q'. Ray 4, starting parallel to the axis, is reflected as if it came from F. 
Ray 6 toward the center of curvature C is reflected back on itself, while 
ray 7 going toward F is reflected parallel to the axis. Since the rays 
never pass through Q', the image Q'M' in this case is virtual. 




Fig. QE. Parallel-ray method for graphically locating the image formed by a concave 




Fig. 6F. Parallel-ray method for graphically locating the image formed by a convex 
mirror. 



The oblique-ray method may also be used for mirrors, as is illustrated 
in Fig. GG for a concave mirror. After drawing the axis 1 and the mirror 
2, we lay out the points C and F and draw a ray 3 making any arbitrary 
angle with the axis. Through F, the broken line 4 is then drawn parallel 
to 3. Where this line intersects the mirror at S, a parallel ray 6 is 
drawn backward to intersect the focal plane at P. Ray 7 is then drawn 
through TP and intersects the axis at M'. By this construction M and 
M' are conjugate points, and 3 and 7 are the parts of the ray in object 
and image spaces. The principle involved in this construction is obvious 



86 



GEOMETRICAL OPTICS 



from the fact that if 3 and 4 were parallel incident rays they would come 
to a focus at P in the focal plane. If in place of ray 4 another ray were 
drawn through C and parallel to ray 3, it too would cross the focal plane 
at P. A ray through the center of curvature would be reflected directly 
back upon itself. 




Fig. 6G. Oblique-ray method for locating the image formed by a concave mirror. 

6.3. Mirror Formulas. In order to be able to apply the standard lens 
formulas of the preceding chapters to spherical mirrors with as little 
change as possible, we must adhere to the following sign conventions: 

1. Distances measured from left to right are positive, while those measured 
from right to left are negative. 

2. Incident rays travel from left to right and reflected rays from right to 
left. 

3. The focal length is measured from the focal point to the vertex. This 
gives / a positive sign for concave mirrors and a negative sign for 
convex mirrors. 

4. The radius is measured from the vertex to the center of curvature. This 
makes r negative for concave mirrors and positive for convex 
mirrors. 

5. Object distances s and image distances s' are measured from the object 
and from the image respectively to the vertex. This makes both s and 
s' positive and the object and image real when they lie to the left 
of the vertex, while they are negative and virtual when they lie to 
the right. 

The last of these sign conventions implies that for mirrors the object 
space and the image space coincide completely, the actual rays of light 
always lying in the space to the left of the mirror. Since the refractive 
index of the image space is the same as that of the object space, the n' 
of the previous equations becomes numerically equal to n. 



SPHERICAL MIRRORS 87 

The following is a simple derivation of the formula giving the conjugate 
relations for a mirror. Referring to Fig. 6G it is observed that by the 
law of reflection the radius CT bisects the angle MTM'. Using a well- 
known geometrical theorem, we may then write the proportion 

MC CM' 



MT M'T 

Now, for paraxial rays, MT ~ MA = s and M'T ~ M'A = s', where 
the symbol ~ means "is approximately equal to." Also, from the 
diagram, 

M C = MA - CA = s + r 
and CM' = CA - M'A = -r - s' = ~(s' + r) 

Substituting in the above proportion, 

s + r s' + r 

8 8' 

which may easily be put in the form 
1 , ! 2 

- + - = MIRROR FORMULA (66) 

S S T 

The primary focal point is defined as that axial object point for which 
the image is formed at infinity, so substituting 8 = f and s' = °o in 
Eq. 66 we have 





/ + 00 


_ 2 

r 




1 

/ 


2 

or 

r 


f = 


r 
2 



from which -.= or / = — - (6c) 

J r l 

The secondary focal point is denned as the image point of an infinitely 
distant object point. This is, s' = /' and s = co , so that 





00 + f 


_ 2 
r 




1 


2 

or 

r 


/' = 


r 
~ 2 



from which - = - _ r /' = - - • (6d) 

Therefore the primary and secondary focal points fall together, and the 
magnitude of the focal length is one-half the radius of curvature. When 
—r/2 is replaced by 1//, Eq. 66 becomes 



just as for lenses. 



l + hj ™ 



88 GEOMETRICAL OPTICS 

The lateral magnification of the image from a mirror may be evaluated 
from the geometry of Fig. 6C. From the proportionality of sides in the 
similar triangles Q' AM' and QAM, we find that -y'/y = s'/s, giving 

m - y - = - - (6/) 

y s 

Example: An object 2 cm high is situated 10 cm in front of a concave 
mirror of radius 16 cm. Find (a) the focal length of the mirror, (b) the 
position of the image, and (c) the lateral magnification. 

Solution: (a) By Eq. 6c, 

-16 rt 



(b) By Eq. 6e, 

giving 

(c) By Eq. 6/, 



10 ^ s' 





2 






1 

s 


or 


1 1 

s' 8 " 


1 1 
" 10 40 




s' = 40 


cm 





m = -n = "4 



The image occurs 40 cm to the left of the mirror, is four times the size of 
the object, and is real and inverted. 

6.4. Power of Mirrors. The power notation that was used in Sec. 4.12 
to describe the image-forming properties of lenses may be readily extended 
to spherical mirrors as follows. As definitions, we let 

P = J V = \ V ' = 7' K = ) (6&) 

Equations 66, 6e, 6c, and 6/ then take the forms 

V + V = -2K (6h) 

V + V = P (6i) 

p = -2K m 

m = y -=-y; «*) 

Example: An object is located 20 cm in front of a convex mirror of 
radius 50 cm. Calculate (a) the power of the mirror, (6) the position of 
the image, and (c) its magnification. 

Solution: Expressing all distances in meters, we have 

K = oM = + 2D and 7 = O20= +5D 

By Eq. 6j, 

P = -2K = -4D Ans. (a) 



SPHERICAL MIRRORS 



89 



By Eq. 6i, 






5 + r = 




1 1 


or s = 


— ■ ^s — : — 




V 9 


By Eq. 6A;, 





m = — 



-4 or 7' = -9 D 
-0.111 m = -11.1 cm Ans.(b) 



-fg = +0.555 Ans. (c) 



The power P = — 4 D, and the image is virtual and erect. It is located 
11.1 cm to the right of the mirror, and has a magnification of 0.555 X. 

6.5. Thick Mirrors. The term thick mirror is applied to a lens system 
in which one of the spherical surfaces is a reflector. Under these circum- 
stances the light passing through the system is reflected by the mirror 




1 — ! 

Fig. QH. Diagrams of several types of "thick mirrors." showing the location of their 
respective focal points. 

back through the lens system, from which it emerges finally into the 
space from which it entered the lens. Three common forms of optical 
systems that may be classified as thick mirrors are shown in Fig. QH. 
In each case the surface furthest to the right has been drawn with a 
heavier line than the others, thereby designating the reflecting surface. 
A parallel incident ray is also traced through each system to where it 
crosses the axis, thus locating the focal point. 

In addition to a focal point and focal plane every thick mirror has a 
principal point and a principal plane. Two graphical methods by which 
principal points and planes may be located are given below. The oblique- 
ray method is applied to (a) the thin lens and mirror combination in Fig. 
67, while the auxiliary-diagram method is applied to (6) the thick lens 
and mirror combination in Fig. 6J. 

In the first illustration the lens is considered thin so that its own prin- 
cipal points may be assumed to coincide at Hi, its center. An incident 
ray parallel to the axis is refracted by the lens, reflected by the mirror, 
and again refracted by the lens before it crosses the axis of the system 
at F. The point T where the incident and final rays, when extended, 



90 



GEOMETRICAL OPTICS 



cross each other locates the principal plane and H represents the prin- 
cipal point. If we follow the sign conventions for a single mirror (Sec. 
6.3), the focal length / of this particular combination is positive and is 
given by the interval FH. 




Fig. 67. Oblique-ray construction for locating the focal point and principal point of a 
thick mirror. 




Fig. 6J. Auxiliary-diagram method of graphically locating the focal point and principal 
point of a thick mirror. 

In the second illustration (Fig. Q>J) the incident ray is refracted by the 
first surface, reflected by the second, and finally refracted a second time 
by the first surface to a point F where it crosses the axis. The point T 
where the incident and final rays intersect locates the principal plane and 
principal point H. 

The graphical ray-tracing construction for this case, shown in the aux- 
iliary diagram in Fig. GJ, is started by drawing XZ parallel to the axis. 



SPHERICAL MIRRORS 91 

With the origin near the center, intervals proportional to n and n' are 
measured off in both directions along XZ. After the vertical lines rep- 
resenting n and n' are drawn the remaining lines are drawn in the order 
of the numbers 1, 2, 3, . . . . Each even-numbered line is drawn paral- 
lel to its preceding odd-numbered line. The proof that this construction 
is exact for paraxial rays is similar to that given for Fig. 3/ in Chap. 3. 
6.6. Thick-mirror Formulas. These formulas will be given in the 
power notation for case (a) shown in Fig. QH. Calling r lt r 2 , and r 3 the 
radii of the three surfaces consecutively from left to right, the power of 
the combination can be shown* to be given by 

P = (1 - cP 1 )(2P 1 + P 2 - cPiPJ (6Z) 

where, for the case in diagram (a) only, and n" = n, 

p 1 = ( n > _ n)(Kx - K t ) (6m) 

P 2 = -2nK 3 (fin) 

and Ki = - K 2 = - K 3 - - 

T\ r 2 r 3 

(see Eqs. 4p and Qd). Of the refractive indices, n' represents that of the 
lens and n that of the surrounding space. The distance from the lens 
to the principal point of the combination is given by 

where c = - (6p) 

n 

It is important to note from Eq. 6o that the position of H is independent 
of the power P 2 of the mirror and therefore of its curvature K 3 . 

Example: A thick mirror like that shown in Fig. QH(a) has as one 
component a thin lens of index n' = 1.50 and radii r t = +50 cm, 
r 2 = —50 cm. This lens is situated 10 cm in front of a mirror of radius 
— 50 cm. Assuming that air surrounds both components, find (a) the 
power of the combination, (6) the focal length, and (c) the principal 
point. 

Solution: By Eq. 6m, the power of the lens is 

P, = (1.50 - 1) Qg - -»„,) - +2 D 

Equation Qn gives for the power of the mirror 

P ° = ~ 2 '^50 = + 4 D 

* For a derivation of these equations, see J. P. C. Southall, " Mirrors, Prisms, and 
Lenses," 3d ed., p. 379, The Macmillan Company, New York, 1936. 



92 GEOMETRICAL OPTICS 

From Eq. 6p, 

d 0.10 „ 1A 
c = - = —J— = 0.10 m 
n 1 

Finally the power of the combination is given by Eq. 61 as 

P = (1 - 0.10 X 2)(2 X 2 + 4 - 0.10 X 2 X 4) 
= 0.8(4 + 4 - 0.8) = 4-5.76 D 

A power of +5.76 D corresponds to a focal length 

f = ~P = 5~76 = °' 173 m = I7,3 Cm 

The position of the principal point 77 is determined from Eq. 6o through 
the distance 

u u °- 10 01 ° A10K 10 - 

HlH = 1 - 0.10 X 2 = O80 = ° 125 m = 12 ° Cm 

It is therefore 12.5 cm to the right of the lens, or 2.5 cm in back of the 
mirror. 

6.7. Other Thick Mirrors. As a second illustration of a thick mirror, 
consider the case of the thick lens silvered on the back as shown in Fig. 
67/(6). A comparison of this system with the one in diagram (a) shows 
that Eqs. 61 to 6p will apply if the powers Pi and P 2 are properly defined. 
For diagram (6), Pi refers to the power of the first surface alone, and P 2 
refers to the power of the second surface as a mirror of radius r 2 in a 
medium of index n'. In other words, 

Pi = P 2 = and c = — (6q) 

With these definitions the power of thick mirror (6) is given by Eq. 61 
and the principal point by Eq. 6o. 

The third illustration of a thick mirror consists of a thin lens silvered 
on the back surface as shown in Fig. 677(c). This system may be looked 
upon (1) as a special case of diagram (a), where the mirror has the same 
radius as the back surface of the thin lens and the spacing d is reduced to 
zero, or (2) as a special case of diagram (6), where the thickness is reduced 
to practically zero. In either case Eq. 6£ reduces to 

P = 2Pi + P 2 (fir) 

and the principal point 77 coincides with 77 x at the common center of the 
lens and mirror. P x represents the power of the thin lens in air and P 2 
the power of the mirror in air, or Pi represents the power of the first sur- 
face of radius r x and P 2 represents the power of the second surface as a 
mirror of radius r 2 in a medium of index n' (see Eq. 65). 



SPHEEICAL MIKRORS 



93 



6.8. Spherical Aberration. The discussion of a single spherical mirror 
in the preceding sections has been confined to paraxial rays. Within this 
rather narrow limitation, sharp images of objects at any distance may be 
formed on a screen, since bundles of parallel rays close to the axis and 
making only small angles with it are brought to a sharp focus in the focal 



Paraxial 
focal 
plane 




Circle 

of least 

confusion 



Fig. 6K. Spherical aberration of a concave spherical mirror. 

plane. If, however, the light is not confined to the paraxial region, all 

rays from one object point do not come to a focus at a common point 

and we have an undesirable effect known as spherical aberration. The 

phenomenon is illustrated in Fig. QK, where parallel incident rays at 

increasing distances h cross the axis closer to the mirror. The envelope 

of all rays forms what is known as 

a caustic surface. If a small screen 

is placed at the paraxial focal plane 

F and then moved toward the 

mirror, a point is reached where the 

size of the circular image spot is a 

minimum. This disklike spot is 

indicated in the diagram and is 

called the circle of least confusion. 

The proof that rays from an outer 
zone of a concave mirror cross the 
axis inside the paraxial focal point 
may be simply given by reference to 
Fig. 6L. According to the law of re- 
flection applied to the ray incident 
at T, the angle of reflection <t>" is equal to the angle of incidence 0. This 
in turn is equal to the angle TCA. Having two equal angles, triangle 
CTX is isosceles, and hence CX = XT. Since a straight line is the 
shortest path between two points, 

CT < CX + XT 




Fig. QL. Geometry showing how marginal 
rays parallel to the axis of a spherical 
mirror cross the axis "inside" the focal 
point. 



94 GEOMETRICAL OPTICS 

Now CT is the radius of the mirror and equals CA, so that 

CA < 2CX 
Therefore £CA < CX 

The geometry of the figure shows that as T is moved toward A, the point 
X approaches F, and in the limit CX = XA = FA = \CA. 

Over the past years numerous methods of reducing spherical aberration 
have been devised. If instead of a spherical surface the mirror form is 
that of a paraboloid of revolution, rays parallel to the axis are all brought 
to a focus at the same point as in Fig. QM (a). Another method is the 




Poraboloidal mirror Mongin mirror 

(a) (6) 

Fig. 6A/. Concave spherical mirrors corrected for spherical aberration. 

one shown later in Fig. 10Q of inserting a "corrector plate" in front of a 
spherical mirror, thereby deviating the rays by the proper amount prior 
to reflection. With the plate located at the center of curvature of the 
mirror, a very useful optical arrangement known as the "Schmidt sys- 
tem" is obtained. Still a third system known as a "Mangin mirror" is 
shown in Fig. QM(b). Here a meniscus lens is employed in which both 
surfaces are spherical. When the back surface is silvered to form the con- 
cave mirror, all parallel rays are brought to a reasonably good focus. 

6.9. Astigmatism. This defect of the image occurs when an object 
point lies some distance from the axis of a concave or convex mirror. 
The incident rays, whether parallel or not, make an appreciable angle 6 
with the mirror axis. The result is that, instead of a point image, two 
mutually perpendicular line images are formed. This effect is known as 
astigmatism and is illustrated by a perspective diagram in Fig. QN. 
Here the incoming rays are parallel, while the reflected rays are converg- 
ing toward two lines S and T. The reflected rays in the vertical or 
tangential plane RASE are seen to cross or to focus at T, while the fan 



SPHERICAL MIRRORS 



93 




Fig. 6jV. Astigmatic images of an off-axis object point at infinity, as formed by a con- 
cave spherical mirror. The lines T and S are perpendicular to each other. 

of rays in the horizontal or sagittal plane JAKE cross or focus at S. If 
a screen is placed at E and moved toward the mirror, the image will 
become a vertical line at S, a circular disk at L, and a horizontal line 
at T. 

If the positions of the T and S 
images of distant object points are 
determined for a wide variety of 
angles, their loci will form a parab- 
oloidal and a plane surface respec- 
tively, as shown in Fig. 60. As the 
obliquity of the rays decreases and 
they approach the axis, the line 
images not only come closer together 
as they approach the paraxial focal 
plane, but they shorten in length. 
The amount of astigmatism for any pencil of rays is given by the distance 
between the T and S surfaces measured along the chief ray. 

Equations giving the two astigmatic image positions are as follows:* 




S' T 

Fig. 60. Astigmatic surfaces for a con- 



cave mirror. 



S 

i + 4- 

8 * S' s 



r cos <f> 
2 cos <f> 



* For a derivation of these equations, see G. S. Monk, "Light, Principles and Experi- 
ments," 1st ed., pp. 52 and 424, McGraw-Hill Book Company, Inc., New York, 1937. 



96 GEOMETRICAL OPTICS 

In both equations s and s' are measured along the chief ray. The angle 
<t> is the angle of obliquity of the chief ray, and r is the radius of curva- 
ture of the mirror. 

The Schmidt optical system, which will be discussed later (Fig. 10Q), 
and the Mangin mirror shown in Fig. GM(b) constitute instruments in 
which the astigmatism of a spherical mirror is reduced to a minimum. 
While the two focal surfaces T and S exist for these devices, they lie very 
close together, and the loci of their mean position (such as L in Fig. 6A/) 
form a nearly spherical surface. The center of this spherical surface 
is located at the center of curvature of the mirror as is shown in 
Fig. 10Q. 

A paraboloidal mirror, while it is free from spherical aberration even 
for large apertures, shows unusually large astigmatic S — T differences 
off the axis. It is for this reason that paraboloidal reflectors are limited 
in their use to devices that require a small angular spread, such as 
astronomical telescopes and searchlights. 

PROBLEMS 

1. The radius of a spherical mirror is —30.0 cm. An object +4.0 cm high is 
located in front of the mirror at a distance of (a) 60.0 cm, (6) 30.0 cm, (c) 15.0 cm, and 
(d) 10.0 cm. Find the image distance for each of these positions. 

2. Solve Prob. 1 graphically. Make separate diagrams for each part. 

Ans. (a) +20 cm. (6) +30 cm. (c) » . (d) -30 cm. 

3. The radius of a spherical mirror is —20.0 cm. An object 2.0 cm high is located 
in front of the mirror at a distance of (a) 30.0 cm, (b) 20.0 cm, (c) 12.0 cm, and (d) 
6.0 cm. Find the image distance for each of these object distances. 

4. Solve Prob. 3 graphically. Make separate diagrams for each part. 

Ans. (a) +15 cm. (6) +20 cm. (c) +60 cm. (d) -15 cm. 

5. The radius of a spherical mirror is +20.0 cm. An object 3.0 cm high is situated 
in front of the mirror at a distance of (a) 30.0 cm, (b) 20.0 cm, (c) 10.0 cm, and (d) 
5.0 cm. Find the image distance for each of these object distances. 

6. Solve Prob. 5 graphically. Make separate diagrams for each part. 

Ans. (a) -7.5 cm. (6) -6.66 cm. (c) -5.0 cm. (d) -3.33 cm. 

7. The radius of a spherical mirror is +12.0 cm. An object 2.0 cm high is located 
in front of the mirror at a distance of (a) 15.0 cm, (6) 10.0 cm, (c) 6.0 cm, and (d) 3.0 
cm. Find the image distance for each of these object distances. 

8. Solve Prob. 7 graphically. Make separate diagrams for each part. 

Ans. (a) -4.28 cm. (6) -3.75 cm. (c) -3.0 cm. (d) -2.0 cm. 

9. A concave mirror is to be used to focus the image of a flower on a nearby wall 
120 cm from the flower. If a lateral magnification of — 16 is desired, what should be 
the radius of curvature of the mirror? 

10. A thin equiconvex lens of index 1.60 and radii 12.0 cm is silvered on one side. 
Find the power of this system for light entering the unsilvered side. .4ns. +36.66 D. 

11. A thin lens of index 1.60 has radii n = +4.0 cm and r. = - 10.0 cm. If the 
second surface is silvered, what is the power of the system? 

12. A thin lens of index 1.75 has as radii ri = —5.0 cm and r 2 = —10.0 cm. If the 
second surface is silvered, what is the power of the system ? Use (a) the special-case 



SPHERICAL MIRRORS 97 

formulas (Kqs. 6q and 6r), and (6) the thick-lens formulas (Eqs. 61, 6wi, and 6n), with 
d = 0. Ans. +5.0 D. 

13. A thin lens with a focal length of + 12.0 cm is located 2.0 cm in front of a spheri- 
cal mirror of radius —20.0 cm. Find (a) the power, {b) the focal length, (c) the princi- 
pal point, and (d) the focal point. 

14. Solve Prob. 13 graphically. Use the method of Fig. 6/. 

Ans. (a) +20.83 D. (b) +4.80 cm. (c) #,// = +2.40 cm. (d) H^F 2.40 cm 

16. A thin lens with a focal length of — 14.5 cm is placed 3.0 cm in front of a spheri- 
cal mirror of radius — 12.5 cm. Find (a) the power, (b) the focal length, (c) the princi- 
pal point, and (d) the focal point. 

16. Solve Prob. 15 graphically. Use the method of Fig. 67. 

Am. (a) +6.65 D. (b) +15.0 cm. (c) H t H = +2.48 cm. (d) H,F = - 12.52 cm. 

17. A thick lens of index 1.60 has radii ft = +12.0 cm and r 2 = —32 cm. If the 
second surface is silvered and the lens is 3.0 cm thick, find (a) the power, (6) the focal 
length, (c) the principal point, and (d) the focal point. 

18. Solve Prob. 17 graphically. Use the met hod of Fig. 6 J. 

Ans. (a) +17.2 D. (b) +5.80 cm. (c) H t ff = +2.07 cm. (d) HxF = -3.73 cm. 

19. A lens 3.68 cm thick, of index 1 .84 and radii r\ = —6.0 cm and r 2 = — 12.0 cm, 
has its second surface silvered as a mirror. Find (a) the power, (6) the focal length, 
(c) the principal point, and (d) the focal point. 

20. Solve Prob. 19 graphically. Use the method of Fig. 6/. 

Ans. (a) +14.4 D. (b) +6.9 cm. (c) Htf = +1.56 cm. (d) H t F = -5.38 cm. 

21. The curved surface of a plano-convex lens has a radius of 12.0 cm. The index 
is 1.60, and the thickness is 3.2 cm. If the curved surface is silvered, find (a) the 
power, (6) the focal length, (c) the principal point, and (rf) the focal point. 

22. Solve Prob. 21 graphically. Use the method of Fig. 6/. 

Ans. (a) +26.7 D. (b) +3.75 cm. (c) H X H = +2.0 cm. (d) H X F = -1.75 cm. 

23. If the plane surface of the lens given in Prob. 21 is silvered in place of the 
curved surface, what are the answers to (a), (6), (c), and (d)? 

24. Solve Prob. 23 graphically. Use the method of Fig. 6J. 

Ans. (a) +9.0 D. (fe) +11.1 cm. (c) HiH = 2.22 cm. (d) HyF = -8.89 cm. 

25. An object is located 15.0 cm in front of a mirror of radius —20.0 cm. Plot a 
graph of the two astigmatic surfaces from = 0° to <t> = 30°. 

26. Plot a graph of the two astigmatic surfaces for a spherical mirror having a radius 
of —16.0 cm. Assume parallel incident light, and show curves from the axis out to 
30°. 



CHAPTER 7 

THE EFFECTS OF STOPS 



There are two subjects in geometrical optics which, though very 
important from a practical standpoint, are frequently neglected because 
they do not directly concern the size, position, and sharpness of the 
image. One of these is the question of the field of view, which determines 
how much of the surface of a broad object can be seen through an optical 
system. The other subject is that of the brightness of images and the 
distinction between this, which is important for visual effects, and the 



Aperture 




Field 



Fig. 7 A. Diagram showing the difference between a field stop and an aperture stop. 

illuminance, which is important for photographic effects. In treating 
both the field of view and the brightness of images it is of primary impor- 
tance to understand how and where the bundle of rays traversing the 
system is limited. The effect of stops or diaphragms, which will always 
exist if only as the rims of lenses or mirrors, must first be investigated. 
7.1. Field Stop and Aperture Stop. In Fig. 1 A a single lens with two 
stops is shown forming the image of a distant object. Three bundles of 
parallel rays from three different points on the object are shown as 
brought to a focus in the focal plane of the lens. It may be seen from 
these bundles that the stop close to the lens limits the size of each bundle 
of rays, while the stop just in front of the focal plane limits the angle at 

98 



THE EFFECTS OF STOPS 



99 



which the incident bundles can get through to this plane. The first is 
called an aperture stop. It obviously determines the amount of light 
reaching any given point in the image and therefore controls the bright- 
ness of the latter. The second, or field stop, determines the extent of the 
object, or the field, that will be represented in the image. 

7.2. Entrance and Exit Pupils. A stop P'E'L' placed behind the lens 
as in Fig. IB is in the image space and limits the image rays. By a 
graphical construction or by the lens formula, the image of this real stop, 




Image 



Fig. IB. Showing how an aperture stop and its image can become the exit and entrance 
pupils, respectively, of a system. 

as formed by the lens, is found to lie at the position PEL shown by the 
broken lines. Since P'E'L' is inside the focal plane, its image PEL lies 
in the object space and is virtual and erect. It is called the entrance 
pupil, while the real aperture P'E'L' is, as we have seen, called the aper- 
ture stop. When it lies in the image space, as it does here, it becomes 
the exit pupil. (For a treatment of object and image spaces see Sec. 4.11.) 
It should be emphasized that P and P', E and E', and L and L' are 
pairs of conjugate points. Any ray in the object space directed through 
one of these points will after refraction pass through its conjugate point 
in the image space. Ray IT directed toward P is refracted through P', 
ray KR directed toward E is refracted through E' , and ray NU directed 
toward L is refracted through L'. The image point Q' is located graph- 
ically by the broken line JQ', parallel to the others and passing unde- 
viated through the optical center A. The aperture stop P'E'L' in the 
position shown also functions to some extent as a field stop, but the 



100 



GEOMETRICAL OPTICA 



edges of the field will not be sharply limited. The diaphragm which 
acts as a field stop is usually made to coincide with a real or virtual 
image, so that the edges will appear sharo. 

7.3. Chief Ray. Any ray in the object space that passes through the 
center of the entrance pupil is called a chief ray. Such a ray after refrac- 
tion also passes through the center of the exit pupil. In any actual 
optical instrument the chief ray rarely passes through the center of any 
lens itself. The points E and E J at which the chief ray crosses the axis 



Exit 

PT' Entrance 
pupil 




Image 

Fig. 7C A front stop and its image can become the respective entrance and exit pupils 
of a system. 

are known as the entrance pupil point and the exit pupil point. The 
former, as we shall see, is particularly important in determining the field 
of view. 

7.4. Front Stop. In certain types of photographic lenses a stop is 
placed close to the lens, either before it (front stop) or behind it (rear stop) . 
One of the functions of such a stop, as will be seen in Chap. 9, is to improve 
the quality of the image formed on the photographic film. With a front 
stop as shown in Fig. 1C, its small size and its location in the object space 
make it the entrance pupil. Its image P'E'U formed by the lens is in 
the image space and constitutes the exit pupil. Parallel rays IT, JW, 
and NU have been drawn through the two edges of the entrance pupil 
and through its center. The lens causes these rays to converge toward 
the screen as though they had come from the conjugate points P', E', 
and L' in the exit pupil. Their intersection at the image point Q' occurs 
where the undeviated ray KA crosses the secondary focal plane. Note 
that the chief ray is directed through the center of the entrance pupil 



THE EFFECTS OF STOPS 



101 



in the object space and emerges from the lens as though it had come 
from the center of the exit pupil in the image space. 

While a certain stop of an optical system may limit the rays getting 
through the system from one object point, it may not be the aperture 
stop for other object points at different distances away along the axis. 
For example, in Fig. ID a lens with a front stop is shown with an object 
point at M . For this point the periphery of the lens itself becomes the 
aperture stop, and since it limits the object rays it is the entrance pupil. 




L J stop 
Image 
Fig, ID. The entrance and exit pupils are not the same for all object and image points. 

Its image, which is again the lens periphery, is also the exit pupil. The 
lens margin is therefore the aperture stop, the entrance pupil, and the 
exit pupil for the point M. If this object point were to lie to the left 
of Z, PEL would become the entrance pupil and the aperture stop, and 
its image P'E'U the exit pupil. 

In the preliminary design of an optical instrument it may not be known 
which element of the system will constitute the aperture stop. As a 
result the marginal rays for each element must be investigated one after 
the other to determine which one actually does the limiting. Regardless 
of the number of elements the system possesses, it will usually be found 
to contain but one limiting aperture stop. Once this stop is located, 
the entrance pupil of the entire system is the image of the aperture stop 
formed by all lenses preceding it and the exit pupil is the image formed by 
all lenses following it. Figures IB and 1C, where there is only a single 
lens either before or behind the stop, should be studied in connection 
with this statement. 

7.6. Stop between Two Lenses. A common arrangement in photo- 
graphic lenses is to have two separate lens elements with a variable stop 
or iris diaphragm between them. Figure IE is a diagram representing 
such a combination, and in it the elements (1) and (2) are thin lenses 
while P E L is the stop. By definition the entrance pupil of this system 



102 



GEOMETRICAL OPTICS 



is the image of the stop formed by lens (1). This image is virtual, erect, 
and located at PEL. Similarly by definition the exit pupil of the entire 
system is the image of the stop formed by lens (2) . This image, located 
at P'E'L', is also virtual and erect. The entrance pupil PEL lies in the 
object space of lens (1), the stop P E L lies in the image space of lens (1) 
as well as in the object space of lens (2), and the exit pupil P'E'L' lies 
in the image space of lens (2). Points Po and P, E and E, and L and L 
are conjugate pairs of points for the first lens, while P and P', E and 
E', and L and L' are conjugate pairs for the second lens. This makes 



Entrance 
pupil 




Fig. IE, Stop between two lenses. The entrance pupil of a system is in its object 
space, while the exit pupil is in its image space. 

points like P and P' conjugate for the whole system. If a point object 
is located on the axis at M, rays MP and ML limit the bundle that will 
get through the system. At the first lens these rays are refracted through 
Po and Lo, and at the second lens they are again refracted in such direc- 
tions that they appear to come from P' and L' as shown. The purpose 
of using primed and unprimed symbols to designate exit and entrance 
pupils respectively should now be clear; one lies in the image space, the 
other in the object space, and they are conjugate images. 

The same optical system is shown again in Fig. IF for the purpose of 
illustrating the path of a chief ray. Of the many rays that can start 
from any specified object point Q and traverse the entire system, a chief 
ray is one which approaches the lens in the direction of E, the entrance 
pupil point, is refracted through E , and finally emerges traveling toward 
Q' as though it came from E', the exit pupil point. 

7.6. Two Lenses with No Stop. The theory of stops is applicable not 
only to cases where circular diaphragms are introduced into an optical 
system but to any system whatever, since actually the periphery of any 



THE EFFECTS OF STOPS 



103 



lens in the system is a potential stop. In Fig. 1G two lenses (1) and (2) 
are shown, along with their mutual images as possible stops. Assuming 
Pi to be a stop in the object space, its image P' formed by lens (2) lies 
in the final image space. Looking upon P 2 as a stop in the image space, 



Entrance 
pupil 




Fig. IF. The direction taken by any chief ray is such that it passes through the centers 
of the entrance pupil, the stop, and the exit pupil. 



(1) 



(2) 



¥ 



n 



4 



+^i 



Axis 



*x 



E' 



E F[ 



f; 



i: 

T 

L 
T 
I 
I 
Fig. 1G. The margin of any lens may be the aperture stop of the system. 



its image P formed by lens (1) lies in the first object space. There are 
therefore two possible entrance pupils, Pi and P, in the object space of 
the combination of lenses, and two possible exit pupils, P 2 and P', in 
the image space of the combination. For any axial point M lying to the 
left of Z, Pi becomes the limiting stop and therefore the entrance pupil 



104 



GEOMETRICAL OPTICS 



of the system. Its image P' becomes the exit pupil. If, on the other 
hand, M lies to the right of Z, P becomes the entrance pupil and Pi 
the exit pupil. 

7.7. Determination of the Aperture Stop. In the system of two lenses 
with a stop between them represented in Figs. IE and IF, the lenses were 
made sufficiently large so that they did not become aperture stops. If, 
however, they are not large compared with the stop, as may well be the 
case with a camera lens when the iris diaphragm is wide open, the system 
of stops and pupils may become similar to those shown in Fig. 7H. This 




(1) (0) (2) 
Fig. 7H. A system composed of several elements has a number of possible stops and 
pupils. 

system consists of two lenses and a stop, each one of which, along with 
its various images, is a potential aperture stop. P[ is the virtual image 
of the first lens formed by lens (2), P' the virtual image of the stop P 
formed by lens (2), P the virtual image of P formed by lens (1), and 
P 2 the virtual image of the second lens formed by lens (1). In other 
words, when looking through the system from the left one would see the 
first lens, the stop, and the second lens in the apparent positions P h P , 
and Pi. Looking from the right, one would see them at P[, P' Q , and P' 2 . 
Of all these stops Po, Pi, and P 2 are potential entrance pupils located in 
the object space of the system. 

For all axial object points lying to the left of X, Pi limits the entering 
bundle of rays to the smallest angle and hence constitutes the entrance 
pupil of the system. In general the object of which it is the image will 
be the aperture stop, which in this case is the aperture Pi of lens (1) 
itself. The image of the entrance pupil formed by the entire lens system, 
namely P[, constitutes the exit pupil. For object points lying between 



THE EFFECTS OF STOPS 



105 



X and Z, P becomes the entrance pupil, P the aperture stop, and P' the 
exit pupil. Finally, for points to the right of Z, Pi is the entrance pupil, 
while P' 2 is both the aperture stop and the exit pupil. It is apparent 
from this discussion that the aperture stop of any system may change 
with a change in the object position. The general rule is that the aperture 
stop of the system is determined by that stop or image of a stop which subtends 
the smallest angle as seen from the object point. If it is determined by an 
image, the aperture stop itself is the corresponding object. In most 
actual optical instruments the effective stop does not change over the 
range of object positions normally covered by the instrument when in use. 

Having established the methods of determining the positions of the 
aperture stop and of the entrance and exit pupils, we may now take up 
the two important properties of an optical system, field of view and 
brightness. To begin with, let us consider the former property. 

7.8. Field of View. When one looks out at a landscape through a 
window, the field of view outside is limited by the size of the window 




Fig. 71. Field of view through a window. 

and by the position of the observer. In Fig. 11 the eye of the observer 
is shown at E, the window opening at JK, and the observed field at GIL 
In this simple illustration the window is the field stop (Sec. 7.1). When 
the eye is moved closer to the window the angular field a is widened, 
while when it is moved farther away the field is narrowed. It is common 
practice with optical instruments to specify the field of view in terms of 
the angle a and to express this angle in degrees. The angle 6 which the 
extreme rays entering the system make with the axis is called the half- 
field angle, and limits the width of the object that can be seen. This 
object field includes the angle 20, and in this instance is the same as the 
image field, of angular width a. 

7.9. Field of a Plane Mirror. The field of view afforded by a plane 
mirror is very similar to that of a simple window. As shown in Fig. 7J, 
TU represents a plane mirror, and P'E'V the pupil of the observer's eye, 
which here constitutes the exit pupil. The entrance pupil PEL is the 
virtual image of the eye pupil formed by the mirror, and is located just 
as far behind the mirror as the actual pupil is in front of it. The chief 
rays E'T and E'U limit the field of view in image space, while the corre- 



106 



GEOMETRICAL OPTICS 



sponding incident rays ER and ES define the field of view in object 
space. The latter show the limits of the field in which an object can be 
situated and still be visible to the eye. In this case also, although not in 
general, it subtends the same angle as does the image field. 

The formation of the image of an object point Q within this field is also 
illustrated. From this point three rays have been drawn toward the 
points P, E, and L in the entrance pupil. Where these rays encounter 
the mirror, the reflected rays are drawn toward the conjugate points 
P', E', and U in the exit pupil. The object Q and the entrance pupil 



Exit B /^ 
pupil 




Entrance 
pupil 



Fig. 7J. Field of view of a plane mirror. 

PEL are in the object space, while the image Q' and the exit pupil P'E'L' 
are in the image space. If Q happens to be located close to RT, only 
part of the bundle of rays defined by the entrance pupil will be intercepted 
by the mirror and will be reflected into the exit pupil. In defining the 
field of view it is customary to use the chief ray RTE', although in the 
present case this distinction is not important because of the relative 
smallness of the pupil of the eye. Its size is obviously greatly exagger- 
ated in the diagram. 

Since the limiting chief ray is directed toward the entrance pupil point 
E, the half-field angle 6 is in general determined by the smallest angle 
subtended at E by any stop, or image of a stop, in the object space. The 
stop determined in this way is the field stop of the system. For a single 
mirror the field stop is the border of the mirror itself. 

7.10. Field of a Convex Mirror. When the mirror has a curvature the 
situation is little changed except that the object field and the image field 
no longer subtend the same angle (0 ^ 6' in Fig. IK). In this figure 
P'E'L' represents the real pupil of an eye placed on the axis of a convex 
mirror TU. The mirror forms an image PEL of this exit pupil, and this 



THE EFFECTS OF STOPS 



107 



is the entrance pupil which is now smaller in size. Following the same 
procedure as for a plane mirror, the lines limiting the image field and the 
object field have been drawn. Rays emanating from an object point Q 
toward P, E, and L of the entrance pupil are shown as reflected towards 



Exit E >^ 
pupil * 




-j? Entrance 
X & pupil 



/ / stop 
Fig. IK. Field of view of a convex mirror. 




Aperture 
stop 



Fig. 7L. Field of view of a converging lens. 



P', E', and L' in the exit pupil. When extended backward these rays 
locate the virtual image Q'. The half-field angle 6 is here larger than 
6', which determines the field of view to the eye. A similar but somewhat 
more complicated diagram can be drawn for the field of view of a concave 
mirror. This case will be left as an exercise for the student, since it is 
very similar to that of a converging lens to be discussed next. 

7.11. Field of a Positive Lens. The method of determining the half- 
field angles 6 and 6' for a single converging lens is shown in Fig. 7L. The 
pupil of the eye, as an exit pupil, is situated on the right, and its real 



108 



GEOMETRICAL OPTICS 



inverted image appears at the left. The chief rays through the entrance 
pupil point E which are incident at the periphery of the lens are refracted 
through the conjugate point E'. 

The shaded areas, or rather cones, ETU and ERS mark the boundaries 
within which any object must lie in order to be seen in the image field. 
The field stop in this case is the lens TU itself, since it determines the 
half-field angle subtended at the entrance pupil point. If the eye, and 
therefore the exit pupil, is moved closer to the lens, thereby increasing 
the image-field angle 6', the inverted entrance pupil moves to the left, 
causing a lengthening of the object-field cone ETU. 



Entrance 
pupil 




, Exit 
pupil 



Fig. 1M. Image formation within the field of a converging lens system. 



The same lens has been redrawn in Fig. 1M, where an object QM is 
shown in a position inside the primary focal point. Through each of the 
three points P, E, and L, rays are drawn from Q to the lens. From there 
the refracted rays are directed through the corresponding points P', E', 
and L' on the exit pupil. Extending them backward to their common 
intersection, the virtual image is located at Q'. The oblique-ray or 
parallel-ray methods of construction (not shown) may be used to con- 
firm this position of the image. It will be noted that if objects are to be 
placed near the entrance pupil point E, they must be very small; other- 
wise only a part of them will be visible to an eye placed at E'. The 
student will find it instructive to select object points that lie outside the 
object field and to trace graphically the rays from them through the lens. 
It will be found that invariably they miss the exit pupil. 

When a converging lens is used as a magnifier, the eye should be placed 
close to the lens, since this widens the image-field angle and extends the 
object field so that the position of the object is less critical. 

7.12. Photometric Brightness and Illuminance. The amount of light 
flowing out from a point source Q within the small solid angle subtended 
by the element of area dA at the distance r [Fig. 7N(a)] is proportional 
to the solid angle. This is found by dividing the area of dA projected 



THE EFFECTS OF STOPS 



109 



normal to the rays by r 2 , so that the luminous flux in this elementary 
pencil may be written 

dA cos <£ 



dF — const. 



(7a) 



Since the source in practice is never a mathematical point, we must con- 
sider all pencils emitted from an element of area dS, as shown for three 
of these pencils in part (b) of Fig. 7.V. Assuming that the source is a 
so-called " Lambert's-law radiator," the flux will now be proportional to 
the projected area of dS as well, so that 



dF = const. 



dS dA cos cos <f> 



(76) 



The value of the constant depends only upon the light source, and is 
called its photometric brightness B. 
To distinguish it from the visual 
sensation of brightness, it is usually 
termed the luminance in the tech- 
nical literature, but here we shall use 
the more common name brightness, 
with the understanding that it is 
the photometric quantity that is 
meant. The unit of B is experi- 
mentally defined as one-sixtieth of 
the brightness of a black body at 
the temperature of melting plati- 
num, and is called the candle per square centimeter. 
unit, the flux becomes 




(6) 
Fig. 7 AT. An elementary pencil and an ele- 
mentary beam. 



Expressing B in this 



dF = B 



dS dA cos cos 4> 



lumens 



(7c) 



This is a quantity which must, aside from small losses due to reflection 
and absorption, remain constant for a bundle of rays as it traverses an 
optical system.* 

The illuminance E of a surface is defined as the luminous flux incident 
per unit area, so that 



dE = 



dF _ B co s dS cos <p 
dA ~ ~i*~ 



(7d) 



Illuminance is often expressed in lumens per square meter, or lux. In 

* To be exact, the expression must be multiplied by n 2 in any medium of index n, 
but since the initial and final media are usually the same, this factor rarely needs to 
be taken into account. 



110 



GEOMETRICAL OPTICS 



order to calculate the illuminance at any point due to a source having a 
finite area, we must integrate Eq. 7d over this area: 



E = 



I 



B dS cos 6 cos <{> 



(7e) 



The exact evaluation of this integral is in general difficult, but in most 
cases the source is sufficiently far from the illuminated surface so that 
we may regard both cos <j> and r 2 as constant. In this case 



E = 



cos <i> 



B cos e dS = 



I cos <f> 



(7/) 



where the integral has been designated by /, since it represents what is 
called the luminous intensity of the source. The definition of this, 
then, is 

/ = jJB cos 6 dS (70) 

The quantities F, B, E, and I are the four basic ones that are dealt with 
in the subject of photometry. 

As an example appropriate to the present subject let us calculate the 
illuminance due to a luminous disk 6 cm in diameter on a small surface 
placed normal to the axis of the disk and 20 cm away from it. The 
brightness of the disk will be taken as 2 candles/cm 2 . 




Fig. 10. Illuminance due to a circular disk. 



This problem is illustrated in Fig. 10. The distance from dA to the 
edge of the disk can be calculated to be only 1.1 per cent greater than 
that to the center; hence r may be regarded as constant. Furthermore 
the angles 6 and <j> at which the light leaves the source and strikes the 
surface are small enough so that we may set cos = cos </> = 1. Equa- 
tion 7e may then be written 



THE EFFECTS OF STOPS 111 

E = ~ [J B dS = y 2 B X2t P dp (7ft) 

since the area dS of an annular element of radius p and width dp is its 
circumference 2irp multiplied by its width dp. But from the figure 

p = r sin a dp = r cos a da 

where a is the half angle subtended by the element. Making these sub- 
stitutions in Eq. 7k, one finds 

1 f"" 
E = -= / 2irr 2 B sin a cos a da 



r' Jo 



= 2-kB 



sin 2 a 



= irB sin 2 a (7i) 



It is instructive to express a in terms of linear dimensions by using the 
relation sin 2 a = po 2 /(po 2 + r 2 ), giving 

„ _ xgpp 2 BS (7 ,, 

Po 2 + r 2 po 2 + r 2 w ' 

Substituting the numerical values, we have 

E " Ww " 5™ " 01382 lume "/ cm ' " 13821ux 

This is not very different from the result that one would obtain by treat- 
ing the source as a point source of luminous intensity / = BS. Using 
Eq. 7/, this would yield 

I cos <p 2X28.27X1 ft1i1Jl , , 

E = ^ = ~n2 = 0.1414 lumen/cm 2 

The condition under which it is legitimate to assume a point source is, 
by Eq. 7j, that p 2 shall be negligible with respect to r 2 . Even if po is 
as large as ^ r , the error is only l per cent. 

7.13. Brightness of an Image. In Fig. 7P is shown a lens forming the 
image dA' of a surface element dS of the object. If the image is observed 
by the eye E, the luminous flux dF entering it is limited by the area 
dA" of the pupil so that only the narrow bundle indicated by the broken 
lines contributes to the image on the retina. Now, since the quantity 
which characterizes a bundle is the multiplier of B in Eq. 7c and since 
this remains constant through the system, we have, neglecting losses, 



dF dS dA cos cos <p _ dS' dA' cos 6' cos <f>' 

J A" ««o a" ««o a." 

(7*0 



B ~ (r) 2 (r') 2 

dS" dA" cos e" cos </>' 



(r") : 



112 



GEOMETRICAL OPTICS 



The last member of this equation refers to the bundle in the region to 
the right of the image, and since we assume the flux in the bundle to 
remain constant and equal to dF, we have 



dF 
B 



dF 
B" 



(71) 



where B" denotes the brightness of the image. Hence the important 
result that 

B" = B (7m) 

For an image formed in the same medium as the object by an optical 
system in which the losses are negligible, the brightness of the image equals 
that of the object. 




— i— i T^ 

Fig. IP. Geometry for treating image brightness. 



This result may seem surprising to one who has experimented at form- 
ing images with a lens, because one always finds that when the image 
is observed on a screen its brightness to the eye increases as the magnifi- 
cation is made smaller. If, however, the image is observed directly by 
the eye, without the use of a screen, its brightness does remain unchanged. 
This is because the brightness represents the flux per unit area per unit 
solid angle, as can be seen from Eqs. 7k and 11 which give, assuming 
cos 6" = cos </>" = 1, 

dF dF 



B" = 



dS 



~ dA' 



dS" da/ 



(7n) 



(r"V 



When the magnification is decreased the flux incident per unit area of the 
image is increased, but the total solid angle a>" (Fig. 7P) is also increased 
in such a way that the brightness stays constant. The light incident 
per unit area on a diffusing screen determines its brightness, but this is 
not the same brightness as the above, since the light is scattered in all 
directions by such a screen. 

7.14. Normal Magnification. In the foregoing discussion, it was 
assumed that the pupil of the eye acts as the aperture stop of the system. 
If this is not the case, for example if in Fig. IP the cone u" emerging 



THE EFFECTS OF STOPS 



113 



from the image is not wide enough to fill the eye pupil, the brightness or 
the image will fall below that of the object. In telescopes and micro- 
scopes the eye is usually placed at the exit pupil of the system, and if 
the full brightness of the object is to be represented in the image, the 
exit pupil must be at least as large as the pupil of the eye. Now the 
diameter of the exit pupil is inversely proportional to the magnification, 
as will be shown for example in the case of a telescope (Eq. 10k). Hence 
the magnification should not exceed that at which the size of the exit 




Fig. 1Q. Illuminance of an image formed by a lens. 

pupil matches that of the eye. This particular value is called the normal 
magnification of the instrument. We shall see that it represents not only 
the maximum allowable value in order to avoid sacrifice of brightness 
but also the minimum value required to realize the full resolving power 
of the instrument (Sec. 15.9). 

7.16. Illuminance of an Image. The illuminance, as defined by 
Eq. 7e, represents the total flux per unit area incident on a surface from 
all directions. It determines the photographic or other energy effects, 
as well as the amount of light scattered by unit area of a diffusing screen. 
To evaluate it in the case of an image formed by a lens or lens system, 
let us represent this system by A in Fig. 1Q, which also shows the posi- 
tions of the entrance pupil PEL and the exit pupil P'E'L'. The bright- 
ness B' of the exit pupil as observed at the image point Q' is equal to 
that of the source, since, from Eq. Ik, 

dF = dS'dA' cosfl' cos <j> ' = dF 
B = (r') 2 = B' 

But the brightness is the flux per unit area per unit solid angle, so that 



114 



GEOMETRICAL OPTICS 



if we wish the total flux incident per unit area we must multiply B' by 
the solid angle a/ subtended by the exit pupil, and this gives 



E = BW = Bu' 



(7o) 



Thus the illuminance of the image is the product of the brightness of the 
source and the solid angle subtended by the exit pupil at the image. This 
relation is not exact, since as may be seen by referring to Eq. 7e, it 
assumes that all angles are small. It is, however, a good approximation 
in most actual cases. As in the previous discussion we are here neglect- 
ing losses by absorption and reflection. The occurrence of co' in Eq. lo 



M' 




Axis 



-Q' 



D 



Fig. 1R. A spotlight or searchlight beam is often rated in terms of its beam candle 
power. 



is the basis for rating the speed of camera lenses by their /-numbers, as 
will be explained in Sec. 10.2. 

It is interesting to note that the illuminance is the same as that which 
would be obtained if the lens were removed and the source were placed 
at the exit pupil and increased in area to the size of the pupil. The result 
of the calculation given in Sec. 7.12 may be used to prove this proposition. 
The illuminance due to a disk of brightness B' , the diameter of which 
subtends a plane angle 2a, was there found to be (see Eq. li) 

E = ttB' sin 2 a 

Provided that a is not too large, a disk of radius r sin a subtends a solid 
angle a/ = (71-r 2 sin 2 a)/r 2 = t sin 2 a, so that 

E = B<*' 

in agreement with Eq. To. 

As a practical illustration of this principle, consider the intense beam 
of light produced by a spotlight or searchlight, as illustrated in Fig. 7R. 
The rim of the reflector of aperture A is the entrance pupil as well as 
the exit pupil. Neglecting losses of light by reflection and absorption, 
the illuminance over the region D on a distant screen M is the same as 



THE EFFECTS OF STOPS 115 

that which would be obtained were the reflector removed and a source 
of the same brightness as S but having the full size of A placed at the 
position of A. The equivalent beam candle power of a spotlight or 
searchlight is defined as the candle power of a bare source which, if 
located at the same distance away from a given point, would produce at 
that point the same illuminance. 

7.16. Image of a Point Source. The above principle is applicable to 
the illuminance of the image of a source of finite area. If the area of 
the source is negligible, as it is for example in the telescopic images of 
stars, the principle deduced above is no longer applicable. The image, 
instead of being of the very small size predicted by geometrical optics, 
is actually broadened because of diffraction by the aperture of the lens 
system (Sec. 1.1). Its illuminance 
is therefore less than would be pre- 
dicted by Eq. lo. The investigation 
of this case requires the results of the 
theory of diffraction and will there- 
fore be postponed until we take up 
this subject (Sec. 15.10). 

7.17. Illuminance off the Axis. 
Supposing the object were a plane 
surface of uniform brightness, it 

would be found that the illuminance ^ 

... ., - „ - ... Fig. IS. Illuminance at an off-axis point 

m the image would fall off with j n tne j magc- 

distance away from the axis. This 

effect is due to more than one cause. In Fig. IS let P'E'U represent the 

exit pupil, which has a uniform brightness B' equal to that of the source. 

At the axial point M' the illuminance is, according to Eq. lo, equal 

to B'o)'. For a point such as Q', however, the following factors act 

to decrease the illuminance: (a) a factor «"/«' = cos 2 0; (b) a factor 

P'L"/P'L' = cos 6, representing the decrease in area of the exit pupil as 

seen from Q' compared with that seen from M'; and (c) another factor 

cos coming from the fact that the light is not incident normally on the 

surface at Q' , as it would be on the surface represented by the broken 

line. Tilting a surface through an angle distributes the flux over an 

area which is 1/cos times larger, and hence the illuminance, or flux 

per unit area, is decreased by cos 0. Putting all these factors together, 

we have, for the illuminance at Q', 

E" = B'a>' cos 4 (7p) 

Near the axis the factor cos 4 varies only slightly from unity, but if a 
becomes as great as 30°, for example, the illuminance is reduced by 
44 per cent. 




116 



GEOMETRICAL OPTICS 



7.18. Vignetting. Another effect, which may cause the illuminance 
off the axis to fall at an even more rapid rate, is that known as vignetting. 
This is particularly likely to occur in a lens system containing stops, as 
is illustrated for a single lens in Fig. IT. Although the aperture of the 
stop is smaller than that of the lens, at the angle of incidence shown some 
of the rays at the top miss the lens entirely, while the lower part of the 
lens receives no light. For distant object points, the field that is repro- 
duced without vignetting covers angles up to the value of a shown in the 




Fig. IT. Illustrating the meaning of the term "vignetting." 

diagram. At wider angles the field begins to darken more rapidly than 
would be indicated by Eq. 7p. Vignetting is seldom encountered in tele- 
scopes or in other instruments having a relatively small field of view 
but in instruments like wide-angle cameras it can become serious. 



PROBLEMS 

1. A thin lens with an aperture of 5.0 cm and a focal length of +4.0 cm has a 3.0-cm 
stop located 2.0 cm in front of it. An object 1.5 cm high is located with its lower end 
on the axis 9.0 cm in front of the lens. Locate graphically and by formula (a) the 
position, and (6) the size of the exit pupil, (c) Locate the image of the object graphi- 
cally by drawing the two marginal rays and the chief ray from the top end of the 
object. 

2. A thin lens with a focal length of +3.0 cm and aperture 4.0 cm has a 2.5-cm 
stop located 1.5 cm in front of it. An object 1.0 cm high is located with its lower 
end on the axis 6.0 cm in front of the stop. Locate graphically and by formula (a) 
the position, and (b) the size of the exit pupil. Locate the image of the object 
graphically by drawing the two marginal rays and the chief ray from the top end of 
the object. Ans. (a) —3.0 cm. (6) +5.0 cm. (c) +5.0 cm. 

3. A thin lens with a focal length of —5.0 cm and aperture 4.0 cm has a 2.0-cm stop 
located 2.0 cm in front of it. An object 4.0 cm high is located with its center on the 
axis 12.0 cm in front of the lens. Find graphically and by formula (a) the position, and 
(6) the size of the exit pupil, (c) Graphically locate the image by drawing the two 
marginal rays and the chief ray from the top end of the object. 

4. A thin lens with a focal length of +5.0 cm and an aperture 6.0 cm has a 3.0-cm 
stop located 3.0 cm behind it. An object 3.0 cm high is located with its center on the 
axis 12.0 cm in front of the lens. Find graphically and by formula (a) the position, 



THE EFFECTS OF STOPS 117 

and (b) the size of the entrance pupil, (c) Locate the image graphically by drawing 
the two marginal rays and the chief ray from the top end of the object. 

Ans. (a) +7.5 cm. (o) 7.5 cm. (c) +8.6 cm. 

5. Two thin lenses with focal lengths of +8.0 cm and +6.0 cm, respectively, and 
with apertures of 5.0 cm, are located 4.0 cm apart. A stop 2.5 cm in diameter is 
located midway between the lenses, and an object 4.0 cm high is located with its 
center 10.0 cm in front of the first lens. Find graphically and by formula (a) the 
position and size of the entrance pupil, and (b) the position and size of the exit pupil, 
(c) Locate the final image by drawing the two marginal rays and the chief ray from the 
top end of the object. 

6. A thin lens L\ with an aperture of 6.0 cm and focal length +6.0 cm is located 4.0 cm 
in front of another thin lens Lt with an aperture of 6.0 cm and focal length —10.0 cm. 
An object 1.0 cm high is located with its center on the axis 18.0 cm in front of L\, 
and a stop 3.0 cm in diameter is located 3.0 cm in front of L\. Calculate the position 
and size of (a) the entrance pupil, (6) the exit pupil, and (c) the image, (d) Solve 
graphically. 

Ans. (a) AiE = -3.0 cm, D n = 3.0 cm. (b) A t E' = -1.0 cm, D x = 3.0 cm. 

(c) AM' = +10.0 cm, 2y' = -1.0 cm. 

7. A thin lens L« with an aperture of 5.0 cm and focal length of +8.0 cm is located 
5.0 cm behind another thin lens L x with an aperture of 6.0 cm and focal length +4.0 cm. 
An object 2.0 cm high is located with its center on the axis 5.0 cm in front of Li, and 
a stop 3.0 cm in diameter is located between the lenses 2.0 cm from L\. Calculate 
the position and size of (a) the entrance pupil, (6) the exit pupil, and (c) the image. 

(d) Solve graphically. 

8. A thin lens L\ with an aperture of 4.0 cm and a focal length of -8.0 cm is located 
3.0 cm in front of another thin lens L 2 with an aperture of 4.0 cm and a focal length of 
+6.0 cm. For light incident on the first lens parallel to the axis calculate the position 
and size of (a) the entrance pupil, and (6) the exit pupil, (c) Solve graphically. 

Ans. (a) AtE = +2.2 cm, D„ = 2.9 cm. (6) AiE' = +3.0 cm, D z = 4.0 cm. 

9. A Coddington magnifier lens (see Fig. 10//) is made from a glass sphere of index 
1.6 and diameter 3.0 cm. The lens aperture is ground to a diameter of 2.0 cm, and 
a groove 0.40 cm deep is ground around its center. Find the position and size of (a) the 
entrance pupil, and (b) the exit pupil. 

10. An exit pupil with a 4.0-cm aperture is located 8.0 cm in front of a spherical 
mirror of radius +20.0 cm. An object 2.0 cm high is centrally located on the axis 
6.0 cm in front of the mirror. Find graphically (a) the entrance pupil, (b) the image, 
and (c) the minimum aperture for the mirror required to see the entire object from 
all points of the exit pupil. 

,4ns. (a) AE = -4.44 cm. (6) AM' = -3.75 cm. (c) 2.13 cm. 

11. An exit pupil 3.0 cm in diameter is located 10.0 cm in front of a spherical mirror 
of +16.0 cm radius. An object 4.0 cm high is centrally located on the axis 6.0 cm 
in front of the mirror. Graphically find (a) the position and size of the entrance pupil. 
(6) Find the position and size of the image by drawing the two marginal rays and 
chief ray from the top of the object. 

12. An exit pupil with a 4.0-cm aperture is located 8.0 cm in front of a mirror of 
+ 12.0 cm radius. An object 5.0 cm high is centrally located on the axis 4.0 cm in 
front of the mirror. Graphically find (a) the position and size of the entrance pupil 
(b) Find the position and size of the image by drawing the two marginal rays and 
chief ray from the top of the object. 

Ans. (a) AE - -3.43cm, D„ = +1.71 cm. {b) AM' = -2A0cm,2y' = +3.0cm. 

13. An exit pupil with a 2.5-cm aperture is located 14.5 cm in front of a mirror 
of —12.5 cm radius. An object 1.5 cm high is centrally placed on the axis 13.0 cm 



118 GEOMETRICAL OPTICS 

in front of the mirror. Graphically find (a) the position and size of the entrance 
pupil. (6) Find the position and size of the image by drawing the two marginal rays 
and the chief ray from the top of the object. 

14. An exit pupil with an aperture of 4.0 cm is located 18.0 cm in front of a mirror 
of —12.0 cm radius. An object 2.0 cm high is centrally located on the axis 14.0 cm 
in front of the mirror, (a) Graphically determine the position and size of the entrance 
pupil, (b) Find the position and size of the image by drawing the marginal rays and 
chief ray from the top of the object. 

Ans. (o) AE = +9.0 cm, D n = -2.0 cm. (6) AM' = + 10.5 cm, 2t/' = -1.50 cm. 

15. Construct to scale a diagram of the object field and image field for a lens with 
an aperture of 4.0 cm and a focal length of +6.0 cm, used as a magnifier. Assume 
the exit pupil to be 2.0 cm wide and located 3.5 cm to the right of the lens, and an 
object 3.0 cm high centrally located 4.0 cm to the left of the lens. Graphically find (a) 
the position and size of the entrance pupil, and (6) the position and size of the image 
by drawing the marginal rays and chief ray from the top of the object. 

16. Make a diagram showing the object field and image field for a lens with an 
aperture of 3.0 cm and a focal length of +4.0 cm, used as a magnifier. Assume the 
exit pupil to be 2.0 cm wide and located 2.5 cm to the right of the lens, and an object 
3.0 cm high centrally located 2.80 cm to the left of the lens, (a) Graphically find the 
position and size of the entrance pupil. (6) Find the position and size of the image 
by drawing the marginal rays and the chief ray from the top of the object. 

Ans. (a) AE = +6.66 cm, D„ - +5.33 cm. (b) AM' = -9.33 cm, 2?/' = +10.0 cm. 

17. The focal length of a thin lens 4.0 cm in diameter is 12.0 cm. If this lens is 
placed midway between the eye and a large object 10.0 cm from the eye, what width 
of the object can be seen through the lens? 

18. Calculate the illuminance in lux due to a frosted lamp bulb of projected area 
50 cm 2 and an average brightness of 2.625 candles/cm* on a surface normal to the 
light and 5.0 m away. (Note: Because of Lambert's law the bulb may be treated as a 
flat surface of the area and brightness given.) Aiis. 5.25 lux 

19. If the lamp in Prob. 17 is displaced 3.0 m in a direction at right angles to the 
original line joining it and the illuminated surface, what will be the new value of the 
illuminance? 

20. A lens with an aperture of 4.0 cm and a focal length of +10.0 cm has a 3.0-cm 
stop located 4.0 cm in front of it. A small disk of brightness 50 candles/cm 2 is 
placed centrally on the axis 20.0 cm from the lens. Calculate (a) the illuminance 
at the image, (6) the size of the exit pupil, and (c) the angle at which vignetting begins. 

Ans. (a) 1.37 lumens/cm* (6) 5.0 cm. (c) 7.1°. 

21. A wide-angle camera lens has a focal length of +12.5 cm and takes photographs 
on a 9.0- by 12.0-cm film. Assuming no vignetting, find the per cent by which the 
exposure is diminished at the corners of the film. 

22. A stop 2.0 cm in diameter is located 2.0 cm in front of a thin lens having a 
diameter of 4.0 cm and a focal length of —10.0 cm. Two centimeters behind this 
lens is another thin lens 6.0 cm in diameter and of focal length +2.5 cm. (a) Find 
graphically the entrance and exit pupils for parallel incident light. (6) An object 
2.0 cm high is located with its lower end on the axis 8.0 cm to the left of the stop. 
Find the image by drawing three rays from the top end of the object through the 
system. Two of these rays are to be the marginal rays, and the other is to be the 
chief ray. (c) Solve (a) by formula, finding the position and size of both pupils. 
(d) Find, by calculation, the position and size of the imape 

Ans. (a) A,E - -2.0 cm from A h D„ = 2.0 cm, A 2 E' =+7 86 cm, D x = 3.58 cm. 
(b) Graph, (c) Same as (a), (rf) A 2 M' = +3.89 cm, 2y' = -0.56 cm. 



CHAPTER 8 

RAY TRACING 



The discussion of image formation by a system of one or more spherical 
surfaces has up to this point been confined to the consideration of paraxial 
rays. With this limitation it has been possible to derive relatively simple 
methods of calculating and constructing the position and size of the 
image. In practice the apertures of most lenses are so large that paraxial 
rays constitute only a very small fraction of all the effective rays. It is 
therefore important to consider what happens to rays that are not par- 
axial. The straightforward method of attacking this problem is to trace 
the paths of the rays through the system, applying Snell's law to the 
refraction at each surface. 

8.1. Oblique Rays. All rays which lie in a plane through the principal 
axis and are not paraxial are called oblique rays. When the law of refrac- 
tion is accurately applied to a number of rays through one or more 
coaxial surfaces, the position of the image point is found to vary with the 
obliquity of the rays. This leads to a blurring of the image known as 
lens aberrations, and the study of these aberrations will be the subject 
of the following chapter. Experience shows that it is possible, by 
properly choosing the radii and positions of spherical refracting surfaces, 
to reduce the aberrations greatly. Only in this way have optical instru- 
ments been designed and constructed having large usable apertures and 
at the same time good image-forming qualities. 

Lens designers follow two general lines of approach to the problem of 
finding the optimum conditions. The first is to use graphical methods 
to find the approximate radii and spacing of the surfaces that should be 
used for the particular problem at hand. The second is to use well- 
known aberration formulas to calculate the approximate shapes and 
spacings. If the results of these methods of approach do not produce 
image-forming systems of sufficiently high quality and better definition 
is required, the method known as ray tracing is applied. The latter 
consists in finding the exact paths of several representative rays through 
the system selected. Some of these rays will be paraxial and some 
oblique, and each is traced from the object to the image. 

If the results are not satisfactory, the surfaces are moved, the radii are 

119 



120 



GEOMETRICAL OPTICS 



changed, and the process is repeated until an apparent minimum of 
aberration is obtained. This is a long and tedious cut-and-try process, 
requiring in some cases hundreds of hours of work. Five-, six-, or seven- 
place logarithms may be required, and certain standard tabular forms 
are printed by the different designers for recording the calculations and 
results (see Table 8-1). Recent researches in electronics have led to the 
development of high-speed calculators capable of ray tracing through 
complicated systems in a very short time. Such calculators undoubtedly 




Fig. SA. A graphical method for ray tracing through a single spherical surface, 
method is exact and obeys Snell's law for all rays. 



The 



are leading to the design and production of new and better high-quality 
optical systems. 

In this chapter we shall first consider the method of graphical ray 
tracing and then the method of calculation ray tracing. Lens aberrations 
and the approximate methods using aberration formulas will be treated 
in Chap. 9. 

8.2. Graphical Method for Ray Tracing. The graphical method for 
ray tracing to be presented here is an extension of the procedure given in 
Sec. 1.3 and shown for refraction at plane surfaces in Figs. \C and 2M. 
It is important to note that while the principles used follow Snell's law 
exactly the accuracy of the results obtained depends upon the precision 
with which the operator makes his drawing. A good drawing board, 
with T square and triangles, or a drafting machine is therefore essential; 
as large a drawing board as is feasible is to be preferred. The use of a 
sharp pencil is a necessity. 

The diagrams in Fig. 8A illustrate the construction for refraction at a 
single spherical surface separating two media of index n and n'. After 



RAY TRACING 



121 



the axis and the surface with a center at C are drawn, any incident ray 
like 1 is selected for tracing. An auxiliary diagram is now constructed 
below, comparable in size, and with its axis parallel to that of the main 
diagram. With the point as a center two circular arcs are drawn 
with radii proportional to the refractive indices. Succeeding steps of the 
construction are carried out in the following order: Line 2 is drawn 
through parallel to ray 1. Line 3 is drawn through points T and C. 
Line 4 is drawn through N parallel to line 3 and extended to where it 





i-^*" " 


^* 




10 
3"A 


4,14] 


_15_ 


- 








^^^7 










^-. 


, *11 N 






c 2 


Axis 


n 




f n \ 




n 




C X Cy 






Fig. 8B. Exact graphical method for ray tracing through a centered system of spherical 
refracting surfaces. 

intersects the arc n' at Q. Line 5 connects and Q, and line 6 is drawn 
through T parallel to line 5. 

In this diagram the radial line TC is normal to the surface at the point 
T and corresponds to the normal NN' in Fig. \C. The proof that such 
construction follows Snell's law exactly is given in Sec. 1.3. 

The graphical method applied to a system involving a series of coaxial 
spherical surfaces is shown in Fig. SB. Two thick lenses having indices 
n' and n", respectively, are surrounded by air of index n = 1.00. In the 
auxiliary diagram below arcs are drawn for the three indices n, n' and 
n" . All lines are drawn in parallel pairs as before and in consecutive 
order starting with the incident light ray 1. Each even-numbered line 
is drawn parallel to the odd-numbered line just preceding it, ending up 
with the final ray 18. Note that the radius of the fourth surface is infinite 
and line 15 drawn toward its center at infinity is parallel to the axis. 
The latter is in keeping with the procedures in Figs. 1C and 2M . 

When the graphical method of ray tracing is applied to a thick mirror, 
the arcs representing the various known indices are drawn on both. 



122 



GEOMETRICAL OPTICS 



sides of the origin as shown in Fig. BC. Again in this case the lines are 
drawn in parallel pairs with each even-numbered line parallel to its 
preceding odd-numbered line. Where the ray is reflected by the concave 
mirror, the rays 10 and 14 must make equal angles with the normal. 
Note that in the auxiliary diagram the corresponding lines 9, 12, and 13 




Fig. 8C Ray tracing through a thick mirror. 



form an isosceles triangle. The particular optical arrangement shown 
here is known as a concentric optical system. The fact that all surfaces 
have a common center of curvature gives rise to some very interesting 
and useful optical properties (see Sec. 10.19). 

8.3. Ray-tracing Formulas. A diagram from which these formulas 
may be derived is given in Fig. 8D. An oblique ray MT making an 
angle with the axis is refracted by the single spherical surface at T so 
that it crosses the axis again at M' . The line TC is the radius of the 
refracting surface and constitutes the normal from which the angles of 



RAY TRACING 



123 



incidence and refraction at T are measured. As regards the signs of the 
angles involved, we consider that 

1. Slope angles are positive when the axis must be rotated counter- 
clockwise through an angle of less than t/2 to bring it into coinci- 
dence with the ray. 

2. Angles of incidence and refraction are positive when the radius of 
the surface must be rotated counterclockwise through an angle of 
less than x/2 to bring it into coincidence with the ray. 

Accordingly, angles 6, <£, and <£' in Fig. &D are positive, while angle 0' 
is negative. 

A\ 




Fig. 8D. Geometry used in deriving the ray-tracing formulas. 

Applying the law of sines to the triangle MTC, one obtains 

sin (ir — 4>) _ sin 
r + s r 

Since the sine of the supplement of an angle equals the sine of the angle 
itself, 

sin sin 



Solving for sin <f>, we find 



r + 



V I Q 

sin 4> = sin 6 



(8a) 



Now by Snell's law the angle of refraction <f>' in terms of the angle of 
incidence <f> is given by 



sin <£' = — ; sin <f> 
n 



(86) 



In the triangle MTM' the sum of all interior angles must equal it. There- 
fore 

e + (x - <t>) + <*>' + (-0') =tt 



124 GEOMETRICAL OPTICS 

which, upon solving for 6', gives 

o' - <b r + e - <t> 



(8c) 



This equation allows us to calculate the slope angle of the refracted ray. 
To find where the ray crosses the axis and the image distance s', the 
Jaw of sines may be applied to the triangle TCM', giving 

— sin 8' sin <j>' 



The image distance is therefore 



s = r — r 



s' - r 



sin o 
sin 0' 



(8d) 



An important special case is that in which the incident ray is parallel to 




Fig. 8E. Geometry for ray tracing with parallel incident light. 



the axis. Under this simplifying condition it may be seen from Fig. SE 
that 



■ jl h 
sin d> = - 

r 



(8c) 



where h is the height of the incident ray PT above the axis. For the 
triangle TCM', the sum of the two interior angles <f>' and 6' equals the 
exterior angle at C. When the angles are assigned their proper signs, 
this gives 

& = *' - 4> (8/) 

The six of the above equations which are numbered form an important 
set by which any oblique ray lying in a meridian plane may be traced 
through a number of coaxial spherical surfaces. A meridian plane is 
defined as any plane containing the axis of the system. While most of the 
rays emanating from an extra-axial object point do not lie in a meridian 



RAY TRACING 



125 



plane, the image-forming properties of an optical system can usually be 
determined from properly chosen meridian rays. Skew rays, or rays that 
are not confined to a meridian plane, do not intersect the axis and are 
difficult to trace. 

8.4. Sample Ray -tracing Calculations. These will be illustrated in the 
case of a double-convex lens with radii r\ = +10 cm and r 2 = —10 cm, 
the lens being made of crown glass having an index n' = 1.52300 for the 
Fraunhofer D fine. If the axial thickness is 2 cm, let us find the focal 




Fig. 8F. Geometry involved in the use of ray-tracing formulas. 



points for parallel rays incident at heights above the axis h = 1.5 cm, 
1.0 cm, 0.5 cm, and cm. 

A diagram for this kind of problem is given in Fig. 8F. Refraction at 
the first surface directs the ray toward the corresponding image point at 
M' . This becomes the object point for the second surface, at which 
the refraction determines the final image point M " . The following two 
tables give the calculations for the refracting surfaces separately. For 
the first surface the incident light is parallel to the axis, and the four 
ray-tracing formulas to be used are Eqs. 86, Sd, 8e, and 8/, namely, 



sin 



h 
r 



sin 4> = — sin 4> 

TV 



and 



0' = 4>' - <f> 



$' 



sin 4>' 
sin 0' 



By substitution of the known values of h and r x in the first equation, 
sin (j> is determined. Inserting this, along with the known n and n', in 
the second equation, we obtain sin <f>'. Having found <f> and 0', we may 
use the third equation to calculate 6'. Finally the values of r h <f>', and 
6' are used in the last equation to obtain the image distance s[. In the 



126 



GEOMETRICAL OPTICS 



use of logarithms for these calculations, the subtraction of one logarithm 
from another to find a quotient is avoided by employing the cologarithms 
of all quantities occurring in the denominator. Thus the operations are 
reduced to those of addition. The procedure is self-evident as regards 
the first three columns of figures in Table 8-1, where it is shown for the first 
refracting surface. 

Table 8-1. Calculations for the First Surface 
rj = +10.0 cm, n' = 1.52300, n = 1.00000 





h = 1.5 cm 


h = 1.0 cm 


h = 0.5 cm 


h = 


log h 

COlOg T\ 


0.176091 
9.000000 


0.000000 
9.000000 


9.698970 
9.000000 




log sin £i 

log n 
colog n' 


9.176091 
0.000000 
9.817300 


9.000000 
0.000000 
9.817300 


8.698970 
0.000000 
9.817300 




log sin (f> l 


8.993391 


8.817300 


8.516270 






5°39'8" 
8°37'37" 


3°45'53" 

5°44'21" 


1°52'53" 
2°51'58" 


0.098490 
0.150000 


0' 


2°58'29" 


1°58'28" 


0°59'8" 


0.051510 


colog sin 0' 

log sin <f>\ 
log ri 


1.284871 
8.993391 
1 . 000000 


1.462764 
8.817300 
1.000000 


1.764463 
8.516270 
1.000000 


1.288108 
8.993391 
1.000000 


log (r, - s',) 


1.278262 


1.280067 


1.280733 


1.281499 


r\ — si 


-18.9785 


-19.0576 


-19.0868 


-19.1205 


i 
«i 


+28.9785 cm 


+29.0576 cm 


+29.0868 


+29.1205 cm 



For the case h = in the right-hand column a special procedure is 
followed. The calculation is started by first finding the number that 
corresponds to one of the values of log sin 4>[ in another column. In the 
present case either column may be used. This number is entered oppo- 
site <t>[ in the table under h = 0. For example, in the column headed 
h = 1.5 cm, we find log sin $[ = 8.993391, and the number corresponding 
to this logarithm (namely, 0.098490) is entered in the last column. Fol- 
lowing the same procedure for the corresponding angle 4> h we find 
log sin 0i = 9.176091, and the number 0.150000 is shown opposite <£i. 
The difference between these two numbers i3 next entered for 8'. Then 
opposite colog sin 6' is written the cologarithm of the number 0.051510, 
namely, 1.288108. From this point on, the procedure is the same as for 



RAY TRACING 



127 



the chosen ray, ft = 1.5, values of log sin <f>[ and log r t being taken from 
the column originally selected. The value of s[ that results will be the 
same whatever auxiliary ray is chosen for the calculation.* 

It is to be noted that the image distance s[ is greatest for h = 0, and 
about one-half of 1 per cent less for the 1.5-cm ray. These slightly dif- 
ferent image points M' become the object points for the second lens 
surface, and the slope angles <t>' of Table 8-1 become the slope angles 0' 
for the incident rays in Table 8-II. Since in the latter case the object 
rays are not parallel to the axis, the four ray-tracing formulas to be used 
for Table 8-II are Eqs. 8a, 86, 8c, and 8d, namely, 

sin 0o = — — sin 0' sin <£ 2 ' = — sin <t>' 2 

and 6' = </> 2 ' + 0' - 4>* r 2 - s' 2 ' = r 2 ^^- 

Starting with the first equation, r 2 is given as — 10.0 cm, and s 2 is the dis- 
tance A 2 M' in Fig. 8F. It is obtained by subtracting from the values 
of s[ in Table 8-1 the lens thickness d = 2.0 cm. 

Taking the ray having ft = 1.5 cm as an example, we have s[ from 
Table 8-1 as 28.9785 cm, which after subtraction of d = 2.0 cm yields 
s 2 = 26.9785 cm. The negative sign signifies that the object ray cor- 
responds to a virtual object. Since both r 2 and s 2 have negative signs, 
the two magnitudes are added in the first equation to give —36.9785 cm. 

In the last column of Table 8-II the value for log sin 6' = 8.711892 is 
obtained from colog sin 0' = 1 .288108 of Table 8-1 . Instead of angles 4> 2 , 
0', and <j>2 in the last column, numbers are obtained by the auxiliary-ray 
method described above in connection with Table 8-1. For example, 
the number 0.291210 corresponds to log sin <£." = 9.464206 and is 
entered opposite <j>" in Table 8-II. The number 0.051510 corresponds to 
log sin 0' = 8.711892 and is entered opposite 0' in the table. Then oppo- 
site colog sin 0" is written 0.819550, which is the cologarithm of the 
number 0.151513. From here on, the procedure corresponds to that for 
the other rays. 

The final figures show that, when parallel rays are incident on the lens 
of Fig. 8F at heights of 1.5 cm, 1.0 cm, 0.5 cm, and cm, the axial inter- 
cepts s 2 ' are at 8.8820 cm, 9.0842 cm, 9.1809 cm, and 9.2202 cm, respec- 
tively. Thus the distance from the lens vertex to the second focal point 
is not a constant but varies slightly for different zones of the lens. This 
defect is called spherical aberration and will be discussed in detail in the 
next chapter. The focal distances s[ and s 2 ' for ft = and for = in 

* The theory of this auxiliary-ray method is set forth in Lummer, "Photographic 
Optics," English translation by S. P. Thompson, p. 126, The Macmillan Company, 
New York, 1900. 



128 



GEOMETRICAL OPTICS 



Table 8-1 1. Calculations fob the Second Surface 
r 2 = -10.0 cm, n" = 1.00000, n' = 1.52300, d = 2.0 cm 



e' 


0' = 2°58'29" 


0' = 1°58'28" 


e' = 0°59'8" 


8' = 


r-i + 4 


-36.9785 


-37.0576 


-37.0868 


-37.1205 


log (fi + .s 2 ) 
colog r 2 
log sin 0' 


1.567949 
9.000000 
8.715129 


1.568877 
9.000000 
8.537233 


1.569219 
9.000000 
8.235537 


1.509614 
9.000000 

8 711892 


log sin <t> 2 

log n' 
colog »" 


9.283078 
0.182700 
0.000000 


9.106110 
0.182700 
0.000000 


8.804756 
0.182700 
. 0000(K) 


9.281506 
0.182700 
0.000000 


log sin <£j' 


9.465778 


9.288810 


8.987456 


9 . 464206 


4 


16°59'37" 

2°58'29" 

11°3'50" 


11°12'32" 

1°58'28" 

7°20'8" 


5°34'31" 

0°59'8" 

3°39'27" 


0.201210 
0.051510 
0.191207 


e" 


8°54'16" 


5°50'53" 


2°54'12" 


0.151513 


colog sin 0" 

log sin <;L 

log r 2 


0.810265 
9.465778 
1.000000 


0.991865 
9.288810 
1.000000 


1.295412 
8.987456 
1.000000 


0.819550 
9.464206 
1 . 000000 


log (r« - s 2 ') 


1.276043 


1.280675 


1.282868 


1.283756 


// 
fi — s 2 


-18.8820 


-19.0842 


-19.1809 


-19.2202 


8 S 


+8.8820 cm 


4-9.0842 cm 


4-9.1809 


+9.2202 cm 


0S 2 


0.3382 cm 


0.1360 cm 


0.0393 cm 






Tables 8-1 and 8-II are identical with the values which would be obtained 
from the paraxial-ray formulas given in Sec. 5.1. 

Whenever a plane surface is encountered, refraction is traced exactly 
by means of Eq. 26. If, for example, the second surface of a lens is 
plane, Snell's law becomes 



and Eq. 26 becomes 



sin 6" = -77 sin 6' 
n" 



,, _ , tan 6' 
Sz ' S2 tiET"' 



where 6" = tft and 6' = <f>' 2 . The calculations are carried out by tabulat- 
ing the proper logarithms as in Table 8-II. 



RAY TRACING 129 



PROBLEMS 



1. A double-convex lens 3 cm thick at the center has radii ft = +14 cm and r 2 = 
— 10 cm and an index of 1.600. If a ray of light parallel to the axis is incident on the 
first surface at a height of 4 cm, apply the graphical method to find the point where 
the ray crosses the axis. 

2. Solve Prob. 1 if the incident ray is at a height of 3.0 cm. Ans. +7.92 cm. 

3. Solve Prob. 1 if the incident ray is at a height of 2.0 cm. 

4. Solve Prob. 1 if the incident ray is at a height of 1.0 cm. Ans. +9.23 cm. 
6. Two glass lenses have the following specifications: 

Lens 1. r t = +10 cm, r 2 = — 12 cm, n' = 1.50, and thickness d = 2.5 cm. 
Lens 2. 1\ = +10 cm, r 2 = +12 cm, n" = 1.70, and thickness d = 2.0 cm. 

These lenses are mounted on a common axis with their nearest vertices 1 cm apart. 
A ray of light, parallel to the axis, is incident on the first lens at a height of 3 cm. 
Apply the graphical method, and find where the final refracted ray crosses the axis. 

6. A thick mirror is composed of a double-concave lens, with radii ri = —11 cm, 
r 2 = +11 cm, n' = 1.50, and thickness 1.5 cm, and a concave mirror of radius —9.5 
cm. The two elements are 3.5 cm apart. A light ray is incident on the first lens 
surface 3.5 cm above, and parallel to, the axis. Apply the graphical method to find 
where the final emergent ray crosses the axis. Ans. 8.5 cm left of first vertex. 

7. A single spherical surface of radius +10.0 cm is ground on the end of a glass rod 
of index 1.65250. Using five-place logarithms, locate the focal points for a parallel 
incident ray at a height of (a) h = 1.5 cm, (b) h = 1.0 cm, (c) h = 0.5 cm, and (d) 
h = cm. 

8. Solve Prob. 7 when r = +6.25 cm. 

Ans. (a) 15.6581, (6) 15.7540, (c) 15.8120, (d) 15.8285 cm. 

9. Solve Prob. 7 when r = +4.00 cm. 

10. Solve Prob. 7 when r = +2.50 cm. 

Ans. (a) 5.8803, (6) 6.1405, (c) 6.2830, (d) 6.3314 cm. 

11. Solve Prob. 7 when r = —10.0 cm. 

12. A lens 1.0 cm thick is made of dense barium crown glass of index 1.62350. It 
has radii r : = +10.0 cm and r 2 = —10.0 cm. Using five-place logarithms, locate 
the focal point for (a) h = 1.5 cm, (6) h = 1.0 cm, (c) 0.50 cm, and (d) h = cm. 

Ans. (a) 7.4907, (6) 7.7011, (c) 7.8220, (d) 7.8623 cm. 

13. Solve Prob. 12 when r, = +6.25 cm and r 2 = -25.0 cm. 

14. Solve Prob. 12 when r\ = +5.00 cm and r 2 = infinity. 

Ans. (a) 7.0154, (6) 7.3056, (c) 7.3965, (d) 7.4033 cm. 

15. Solve Prob. 12 when r t = +2.50 cm and r 2 = —6.25 cm. 

16. Solve Prob. 12 when r t = — 15.0 cm and r* = —3.75 cm. 

Ans. (a) 6.2326, (b) 7.2155, (c) 7.7719, (d) 7.9531 cm. 



CHAPTER 9 

LENS ABERRATIONS 



The processes of ray tracing presented in the last chapter served to 
emphasize the inability of the paraxial-ray formulas of the Gauss theory 
to give an accurate account of image detail. A wide beam of rays 
incident on a lens parallel to the axis, for example, is not brought to a 
focus at a unique point. The resulting image defect is known as spherical 
aberration. The Gaussian formulas developed and used in the preceding 
chapters give, therefore, only an idealized account of the images pro- 
duced with lenses of wide aperture. 

When ray tracing is applied to object points located farther and farther 
off the axis, the observed image defects become more and more pro- 
nounced. The methods of reducing these aberrations to a minimum, and 
thereby permitting the formation of reasonably satisfactory images, 
are one of the chief problems of geometrical optics. It would be impos- 
sible within the scope of this book to give all the details of the extensive 
mathematical theory involved in this problem.* Instead we shall 
attempt to show how most of the aberrations manifest themselves and at 
the same time discuss some of the known formulas to see how they may be 
used in the design of high-quality optical systems. 

9.1. Expansion of the Sine, and First-order Theory. In order to 
formulate a satisfactory theory of lens aberrations, many theoreticians 
have found it convenient to start with the correct and precise ray-tracing 
formulas, as given in Eqs. 8a through 8/, and to expand the sines of each 
angle into a power series. An expansion of the sine of an angle by Mac- 
laurin's theorem gives 

6 a 6 b 6 7 9 

sme = e -h + h.-v + h-- (9a) 

For small angles this is a rapidly converging series. Each member is 
small compared with the preceding member. It shows that for paraxial 
rays where the slope angles are very small we may, to a first approxima- 

* For a more thorough account of lens aberrations the reader is referred to A. E, 
Conrady, "Applied Optics and Optical Design," vol. 1, Oxford University Press, 
New York, 1929. 

130 



LENS ABERRATIONS 131 

tion, neglect all terms beyond the first and write 

sin - 

When is small, the other angles </>. </>', and 0' are also small, provided 
the ray lies close to the axis. By substituting for sin 0, <f> for sin <j>, and 
0' for sin 0', in Eqs. 8a, 86, and 8d, we obtain 



* = r -±?8 



Y = tf + 6 - 4> 



*' - n A, 



s - _ r _ r 



0' 



By the algebraic substitution of the first equation in the second, the 
resultant, equation in the third, and 
this resultant in the fourth, all angles 
may be eliminated. The final equa- 
tion obtained by these substitutions 
is found to be none other than the 
Gaussian formula, 

n n' _ n' — n 

s + 7 r 

This equation and others developed 
from it form the basis of what is 
usually called first-order theory. 

The justification for writing 
sin = 0, etc., for all small angles, is 
illustrated in Fig. 9^4. and in Table 9-1. 
the arc length 6 is only one-half of 1 per cent greater than sin 10°, while 
for 40° it is about 10 per cent greater. These differences are measures 
of spherical aberration and, therefore, of image defects. 

Table 9-1. Values of sin and Its First Three Expansion Terms 




Fig. 9 A. Illustrating the arc of an an- 
gle in relation to its sine. 

For an angle of 10°, for example, 





sin 


e 


3 /3! 


B /5! 


10° 
20° 
30° 
40° 


0.1736482 
0.3420201 
0.5000000 
0.6427876 


0.1745329 
0.3490658 
0.5235988 
0.6981316 


0.0008861 
0.0070888 
0.0239246 
0.0567088 


0.0000135 
0.0000432 
0.0003280 
0.0013829 



9.2. Third-order Theory of Aberrations. If all the sines of angles in 
the ray-tracing formulas (Eq. 8a to 8/) are replaced by the first two terms 
of the series in Eq. 9a, the resultant equations, in whatever form they 
are given, represent the results of third-order theory. Thus sin is replaced 



132 



GEOMETRICAL OPTICS 



by - 3 /3!, sin <f> is replaced by <t> - <t> 3 /3\, etc. The resulting equa- 
tions give a reasonably accurate account of the principal aberrations. 

In this theory the aberration of any ray, i.e., its deviation from the 
path prescribed by the Gauss formulas, is expressed in terms of five 
sums, Si to S i} called the Seidel sums. If a lens were to be free of all 
defects in its ability to form images, all five of these sums would have 
to be equal to zero. No optical system can be made to satisfy all these 
conditions at once. Therefore it is customary to treat each sum sep- 
arately, and the vanishing of certain ones corresponds to the absence 
of certain aberrations. Thus, if for a given axial object point the Seidel 




Fig. 9B. Illustration of the spherical aberration in the image of an axial object point as 
formed by a single spherical refracting surface. 

sum Si = 0, there is no spherical aberration at the corresponding image 
point. If both Si = and #2 = 0, the system will also be free of coma. 
If, in addition to Si = and & 2 = 0, the sums S 3 = and S* = as 
well, the images will be free of astigmatism and curvature of field. If 
finally S 5 could be made to vanish, there would be no distortion of the 
image. These aberrations are also known as the five monochromatic 
aberrations because they exist for any specified color and refractive index. 
Additional image defects occur when the light contains various colors. 
We shall first discuss each of the monochromatic aberrations and then 
take up the chromatic effects. 

9.3. Spherical Aberration of a Single Surface. This is a term intro- 
duced in Sec' 6.8, and shown in Fig. QK, to describe the blurring of the 
image formed when parallel light is incident on a spherical mirror. A 
similar blurring of the image that occurs upon refraction by spherical 
surfaces will now be discussed. In Fig. QB M is an object point on the 
axis of a single spherical refracting surface, and M' is its paraxial image 
point. Oblique rays incident on the surface in a zone of radius h are 
brought to a focus closer to, and at a distance of s' h from, the vertex A. 

The distance N'M', as shown in the diagram, is a measure of the 
longitudinal spherical aberration, and its magnitude is found from the 



LENS ABERRATIONS 



133 



third-order formula, 



n . ri n'-n. \(h 2 n 2 r\(\ . lY/l . n' - n\\ faK . 

Since from the paraxial-ray formula (Eq. 36) we have 

n n' _ n' — n 

the right-hand bracket in Eq. 96 is a measure of the deviations from 
first-order theory. Its magnitude varies with the position of the object 




Fig. 9C Longitudinal spherical aberration for parallel light incident on a single spheri- 
cal refracting surface. 

point and for any fixed point is approximately proportional to h 2 , the square 
of the radius of the zone on the refracting surface through which the rays pass. 
If the object point is at infinity so that the incident rays are parallel 
to the axis as shown in Fig. 9C, this equation reduces to 



n' _ n' h 2 n 2 
4 ~ f + 2/VV 



(9c) 



Again the magnitude of the aberration is proportional to h 2 , the square 
of the height of the ray above the axis. 

9.4. Spherical Aberration of a Thin Lens. The existence of spherical 
aberration for a single spherical surface indicates that it may also occur 
in combinations of such surfaces, as, for example, in a thin lens. Since 
many of the lenses in optical instruments are used to focus parallel 
incident or emergent rays, it is usual for comparison purposes to deter- 
mine the spherical aberration for parallel incident light. Figure 9D(a) 
illustrates this special case and shows the position of the paraxial focal 
point F' as well as the focal points A, B, and C for zones of increasing 
diameter. Diagram (6) in Fig. 9D illustrates the difference between 



134 



GEOMETRICAL OPTICS 




^Paraxial 
focal plane 



~^Lat. S.A. 



i.O 



Fig. 9D. Illustrations of lateral and longitudinal spherical aberration of a lens. 

longitudinal spherical aberration, abbreviated Long. S.A., and lateral 
spherical aberration, abbreviated Lat. S.A. 

As a measure of the actual magnitudes involved in longitudinal spher- 
ical aberration, we may use the focal lengths for three zones of a lens 

which were accurately calculated in 
15 Table 8-II. The results were 9.2202 
cm for paraxial rays, 9.1809 cm for 
rays transversing a zone of radius 
h = 0.5 cm, 9.0842 cm for a zone of 
radius h = 1 cm, and 8.8820 cm for a 
zone of radius h = 1.5 cm. These 
figures give a spherical aberration 
of 0.3382 cm for the 1 .5-cm zone, or 
about 4 per cent of the paraxial focal 
length. A graph showing the varia- 
tion of / with h for tliis lens is given 
in Fig. 9.E. For small h the curve 
approximates to a parabola, and since 
the marginal rays intersect the axis 
to the left of the paraxial focal point, 
the spherical aberration is said to be positive. A similar curve for an 
equiconcave lens would bend over to the right, corresponding to negative 
spherical aberration. 



c 














R" 












A 


V 



0.5 



8.8 8.9 




9.1 9.2 9.3 



9.0 

Fig. 9E. A graph of the variation of 
focal length with ray height h. The dif- 
ferences of / are a measure of spherical 
aberration. 



LENS ABERRATIONS 



135 



A series of positive lenses of the same diameter and paraxial focal 
length but of different shape is presented in Fig. 9F(a). The alteration 
of shape represented in this series is known as bending the lens. Each 



<? = -2.00 



-1.00 



-0.50 



+0.50 



+1.00 



+2.00 



<? = -2.0 



-1.0 



+ 0.5 +1.0 +2.0 




Fig. 9F. (a) Lenses of different shapes but with the same power or focal length. The 
difference is one of bending, (b) Focal length vs. ray height h for these lenses. 

lens is labeled by a number q called its shape factor, defined by the formula 



Q = 



r2 + ri 
rz — ri 



(9d) 



As an example, if the two radii of a converging meniscus lens are 
ri = —15 cm and r 2 — —o cm, it has a shape factor 

-5-15 _ 

9= -5+15 = - 2 

The usual reason for considering the bending of a lens is to find that shape 
for which the spherical aberration is a minimum. That such a minimum 
exists is shown by the graphs of Fig. 9F(b). These curves are drawn for 
the same lenses as shown in (a), and the values were taken from Table 9-II. 
They were calculated by the ray-tracing methods of Chap. 8, Tables 8-1 
and 8-II. It will be noted that lens (5), for which the shape factor 
q is +0.5, has the least spherical aberration. The amount of this aberra- 



136 



GEOMETRICAL OPTICS 



tion for the ray having h = 1 cm is shown for the same series of lenses 
by the curves of Fig. 9G. Over the range of shape factors from about 
q = +0.4 to q = + 1.0 the spherical aberration varies only slightly, since 
it is close to a minimum. At no point, however, does it go to zero. We 
therefore see that by choosing the proper radii for the two surfaces of a 




Fig. 9G. A graph of the spherical aberration for lenses of different shape but the same 
focal length. For the lenses shown h = 1 cm,/ = +10cm,d = 2 cm, and n' = 1.51700. 

lens the spherical aberration can be reduced to a minimum but cannot 
be made to vanish completely. 

Reference to the diagrams of Fig. 9D will show that with spherical sur- 
faces the marginal rays are deviated through too large an angle. Hence 
any reduction of this deviation will improve the sharpness of the image. 
The existence of a condition of minimum deviation in a prism (Sec. 2.8) 
clearly indicates that when the shape of a lens is changed the deviation 
of the marginal rays will be least when they enter the first lens surface 
and leave the second at more or less equal angles. Such an equal division 
of refraction will yield the smallest spherical aberration. For parallel 
light incident on a crown-glass lens, this appears from Fig. 9G to occur 
at a shape factor of about q = +0.7, not greatly different from the plano- 
convex lens, for which q = +1.0. 



LENS ABERRATIONS 137 

Spherical aberration can be completely eliminated for a single lens by 
aspherizing. This is a tedious hand-polishing process by which various 
zones of one or both lens surfaces are given different curvatures. For 
only a few special instruments are such lenses useful enough so that the 
added expense of hand figuring is justified. Furthermore, since it is 
figured for only one object distance, such a lens is not free from spherical 
aberration for other distances. The most common practice in lens design 
is to adhere to the simple spherical surfaces and to reduce the spherical 
aberration by a proper choice of radii. 

9.5. Results of Third-order Theory. Although the derivation of an 
equation for spherical aberration from third-order theory is too lengthy 
to be given here, some of the resulting equations are of interest. For a 
thin lens we have the reasonably simple formula 






8/ 3 n(n - 1) 

[£tti * 2 + 4(n + l)vq + (3n + 2)(n " 1)p2 + iT^l] 



(9e) 



where L„ = —, — —, 



As shown in Fig. 9D(6), s' h is the image distance for an oblique ray travers- 
ing the lens at a distance h from the axis, s' p is the image distance for 
paraxial rays, and / the paraxial focal length. The constant p is called 
the position factor, and q is the shape factor defined by Eq. 9d. The 
position factor is defined as 



(9/) 



P = s^Ts 

Making use of the first-order equation 1// = (1/s) + (I/O, the position 
factor may also be expressed in terms of / as 

p - 2? _ i = i - % (ty) 

r s s 

The difference between the two image distances, s' p — s' h is called the 
longitudinal spherical aberration, here abbreviated Long. S.A. 

Long. S.A. = s' p - s' h 

The intercept of the oblique ray with the paraxial focal plane is the 
lateral spherical aberration and from Fig. 9D(6) is seen to be given by 

Lat. S.A. - («' - s' h ) tan 6' 



138 



GEOMETRICAL OPTICS 



If we solve Eq. 9e for the difference s' — s' h , we obtain 



and 



Long. S.A. = s' p s' h L« 
Lat. S.A. = s' p hL s 



(9/0 



The image distance s' h for any ray through any zone is given by 



sk = 



1 + s'L a 



A comparison of the third-order theory with the exact results of ray trac- 
ing is included in Fig. 9G. When the shape factor is not far from that 
corresponding to the minimum, the agreement is remarkably good. The 
numerical results of third-order theory for the seven lenses of Fig. 9F 
are presented in the last column of Table 9-1 1. 

Table 9-1 1. Spherical Aberration op Lenses Having the Same Focal 

Length but Different Shapes q 

Lens thickness = 1 cm, / = 10 era, n = 1.5000, and h = 1 cm 



Shape of lens 


n 


r 2 


9 


Ray 

tracing 


Third-order 
theory 


1. Concavo-convex 

2. Plano-convex 


-10.000 

00 

20.000 

10.000 

6.666 

5.000 

3.333 


- 3.333 

- 5.000 

- 6.666 
-10.000 
-20.000 

oo 

10.000 


-2.00 
-1.00 
-0.50 


+0.50 
+ 1.00 
+2.00 


0.92 
0.45 
0.26 
0.15 
0.10 
0.11 
0.27 


0.88 
0.43 
0.26 
0.15 


6. Plano-convex 

7. Concavo-convex 


0.10 

0.11 
0.29 



Equations useful in lens design are obtained by finding the shape factor 
that will make Eq. 9e a minimum. This may be done by differentiating 
with respect to the shape factor and equating to zero: 



dq 



S/ 3 



f"2(n 



+ 2)? + 4(n- l)(n + l)p' 



n(n - l) 2 

Equating to zero and solving for q, one obtains 

2(n 2 - \)v 



Q = - 



n + 2 



(9t) 



as the required relation between shape and position factors to produce 
minimum spherical aberration. As a rule a lens is designed for some 
particular pair of object and image distances so that p may be calculated 
from Eq. 9/. For a lens of a given n the shape factor that will produce 
a minimum lateral spherical aberration may be obtained at once from 
Eq. 9i. In order to determine the radii that will correspond to such a 



LENS ABERRATIONS 



139 



calculated shape factor and still yield the proper focal length, one may 
then use the lens makers' formula 

Substitution of values of s, s' and r x , r-> from Eqs. 9g and 9d gives the 
following useful set of equations, due to Coddington: 



s = 



Ti = 



2/ 



1 +V 
2/(n - 1) 
<7 + l 



1 - p 



r 2 = 



2 /(n - 1) 
g-1 



The last two relations give the radii in terms of q and /. 
of these by the other gives 

r J - 9 ~ 1 
r 2 q + 1 



(9;) 



Division of one 



(9fc) 



As a problem let us suppose that a single lens is to be made with a focal 
length of 10 cm and that we wish to find the radii of the surfaces which 
will give the minimum spherical aberration for parallel incident light. 
For simplicity we shall assume that the glass has an index n = 1.50. 
In using Eq. Qi the position factor p and the shape factor q must first 
be determined. Substitution of s = oo and s' = 10 cm in Eq. 9/ gives 

10 - oo 

It may be seen that if s is not infinite but is allowed to approach infinity, 
the ratio (s' + s):(s' — s) will approach the value — 1, and will in the 
limit be equal to this. Substituting this position factor in Eq. 9i, we 
obtain 



Q = 



2(2.25 -!)(-!) 
1.5 -1-2 



- a = °- 714 



This value falls at the minimum, of the curve of Fig. 9G. The ratio of 
the two radii is given by Eq. 9k as 






0.714 - 1 
0.714 + 1 



-0.286 
1.714 



= -0.167 



The negative sign means that the surfaces curve in opposite directions, 
and the numerical value indicates a ratio of the radii of about 6:1. Their 
individual values are found from Eq. 9j to be 



ri = 



10 
1.714 



= 5.83 cm 



and 



7*2 = 



10 
0.286 



= -35.0 cm 



140 GEOMETRICAL OPTICS 

Such a lens lies between lenses (5) and (6) in Fig. 9F and has essen- 
tially the same amount of spherical aberration as either one. For this 
reason plano-convex lenses are often employed in optical instruments 
with the convex side facing the parallel incident rays. Should such a 
lens be turned around so that the flat side is toward the incident light, 
its shape factor becomes q = — 1.0, and the spherical aberration increases 
about fourfold. 

Although spherical aberration cannot be entirely eliminated for a single 
spherical lens, it is possible to do so for a combination of two or more 
lenses of opposite sign. The amount of spherical aberration introduced 
by one lens of such a combination must be equal and opposite to that 
introduced by the other. If for example the doublet is to have a positive 
power and no spherical aberration, the positive lens should have the 
greater power and its shape should be at or near that for minimum 
spherical aberration, while the negative lens should have a smaller power 
and its shape should not be near that for the minimum. Neutralization 
by such an arrangement is possible because spherical aberration varies 
as the cube of the focal length, and therefore changes sign with the sign 
of/ (see Eq. 9e). In a cemented lens of two elements, the two interfaces 
should have the same radius. The other two may then be varied and 
then used to correct for spherical aberration. With four radii to manipu- 
late, other aberrations like chromatic aberration can be reduced at the 
same time. This subject will be considered in Sec. 9.13. 

9.6. Fifth-order Spherical Aberration. The two curves that were 
given in Fig. 9G show that, for a lens having a shape factor anywhere 
near the optimum, the agreement between the exact results of ray tracing 
and the approximate results of third-order theory is remarkably good. 
For larger values of h, however, and for shapes further removed from 
the optimum, appreciable differences occur. This indicates the necessity 
of including the fifth-order terms in the theory. The third-order equa- 
tion 9e shows that spherical aberration should be proportional to h 2 , so 
that the curves in Fig. 9F(b) should be parabolas. Nevertheless accurate 
measurements show that for larger h departures from proportionality to 
h 2 do occur and that spherical aberration is more closely represented by 
an equation of the form 

Long. S.A. = ah 2 + bh* (9Z) 

where a and b are constants. The term ah 2 represents the third-order 
effect and bh* the fifth-order effect. Some numerical results for a single 
lens, indicating the necessity for the inclusion of the latter term, are 
shown in Table 9-III. The boldface values in the fifth row are the true 
values for longitudinal spherical aberration, obtained by ray-tracing 



LENS ABERRATIONS 



141 



methods, while those in the last row correspond to a parabola that has 
been fitted at h = 1.0 cm to the equation 



Long. S.A. = a'h- 



with a' = 0.11530 cm- 1 . 



Tablk 9-III. Fifth-order Correction to Spherical Aberration 
/ = 10 cm, ri = -f-5 cm, r 2 = <*>,n = 1.500, d = 1 cm 



1. h, cm 


0.5 


1.0 


1.5 


2.0 


2.5 


3.0 


2. ah* 

3. bh* 


0.02839 
0.00011 
0.02850 
0.02897 

0.02882 


0.11356 
0.00174 
0.11530 
0.11530 
0.11530 


0.25551 
0.00881 
0.26432 
0.26615 
0.25942 


0.45424 
0.02784 
0.48208 
0.48208 
0.46120 


0.70975 
0.06797 
0.77772 
0.77973 

0.71812 


1.02204 
0.14094 


4. ah 2 +bh* 

5. Ray tracing 

6. Parabola 


1 . 16928 
1.16781 
1.03770 



2cm 



The second row gives the third-order corrections ah 2 and the third row 
the fifth-order corrections bh 4 . The fourth row contains the values 

2.5 

2.0 
1.5 
1.0 

0.5 



















a 


h 2 ' 


























Thi 


rd- 


ord 


er 




\ 

















0.6 



0.4 0.2 
— Long. S.A. 



lo 

0.10 





-^ 


\ 


\ 








b 


\ 


\ 










\ 


F 


ft/ 


-Or 


ler 















0.05 
-Long. S.A. 




9.90 



9.95 10.00 10.05cm 
Focal length 



(a) (6) (c) 

Fig. 97/. (a) Third-order and (6) fifth-order contributions to longitudinal spherical 
aberration, (c) Longitudinal spherical aberration of a corrected doublet as used in 
telescopes. 

calculated from Eq. 9Z by fitting the curve at the two points h = 1 cm 
and h = 2 cm. Assuming the values 0. 1 1530 and 0.48208 at these points, 
the constants become 



a = 0.11356 



and 



b = 0.00174 



A comparison of the totals in the fourth row with the correct values in 
the fifth row reveals the excellent agreement of the latter with Eq. 9/. 
Graphs of the values in rows 2 and 3 are given in Fig. 9H, and show the 



142 GEOMETRICAL OPTICS 

negligible contribution of the fifth-order correction at small values of h. 
If only the third-order aberration were present in a lens it would be pos- 
sible to combine a positive and a negative lens having equal aberrations 
to obtain a combination corrected for all zones. Because they actually 
would have different amounts of fifth-order aberration, however, such a 
combination can be corrected for one zone only. 

A graph illustrating the spherical aberration of a cemented doublet 
which is corrected for the marginal zone is shown in Fig. 9//(c) . It will be 
seen that the curve comes to zero only at the origin and at the margin. 
The combination becomes badly overcorrected if the aperture is further 
increased. The plane of best focus lies a little to the left of the paraxial 
and marginal focal points, and its position (the vertical broken fine) 
corresponds to that of the circle of least confusion. 

Let a and b in Eq. 91 represent the constants for a thin-lens doublet. 
If the combination is to be corrected at the margin, i.e., for a ray at the 
height h m , we must have 

Long. S.A. = ah m 2 + bh m 4 = 
or a = — bh m 2 

Substitution in Eq. 91 yields 

Long. S. A. = -bh m 2 h 2 + bh* 

where h m is fixed and h may take any value between and h m . To find 
where this expression has a maximum value, we differentiate with respect 
to h and equate to zero, as follows: 

d ( lop g- S - A -) = -2bh m 2 h + 46A» = 
ah 

Dividing by —2bh, we obtain 

h = hm VI = 0.707/u 

as the radius of the zone at which the aberration reaches a maximum 
[see Fig. 9H(c)]. In lens design spherical aberration is always investi- 
gated by tracing a ray through the combination for the zone of radius 
0.707/i m . 

9.7. Coma. The second of the monochromatic aberrations of third- 
order theory is called coma. It derives its name from the cometlike 
appearance of the image of a point object located just off the lens axis. 
Although the lens may be corrected for spherical aberration and may 
bring all rays to a good focus on the axis, the quality of the images of 
points just off the axis will not be sharp unless the lens is also corrected 
for coma. Figure 9/ illustrates this lens defect for a single object point 
infinitely distant and off the axis. Of the fan of rays in the meridian 



LENS ABERRATIONS 



143 



plane that is shown, only those through the center of the lens form an 
image at A'. Two rays through the margin come together at B' . Thus 
it appears that the magnification is different for different parts of the 




Tangential 
section 



Fig. 9/. Illustrating coma, the second of the five monochromatic aberrations of a lens. 
Only the tangential fan of rays is shown. 

lens. If the magnification for the outer rays through a lens is greater 
than that for the central rays, the coma is said to be positive, while if the 
reverse is true as in the diagram, the coma is said to be negative. 

The shape of the image of an off -axis object point is shown at the upper 
right in Fig. 91. Each of the cir- 
cles represents an image from a dif- 
ferent zone of the lens. Details of 
the formation of the comatic circle 
by the light from one zone of the 
lens are shown in Fig. 9J. Rays 
(1), which correspond to the tan- 
gential rays B in Fig. 91, cross at (1) 
on the comatic circle, while rays 
(3), called the sagittal rays, cross 
at the top of that circle. In gen- 
eral all points on a comatic circle 

are formed by the crossing of pairs of rays passing through two diamet- 
rically opposite points of the same zone. Third-order theory shows that 
the radius of a comatic circle is given by 




Fig. 9J. Each zone of a lens forms a ring- 
shaped image called a comatic circle. 



C, = ^ (Gp + Wq) 



(9m) 



where j, h, and / are the distances indicated in Fig. 9K(a) and p and q 
are the Coddington position and shape factors given by Eqs. 9/ and 9d. 
The other two constants are defined as 



G = 



3(2w + 1) 
4n 



and 



W = 



3(n + 1) 
4n(n — 1) 



144 



GEOMETRICAL OPTICS 





<«> (ft) 

Fig. 9K. Geometry of coma, showing the relative magnitudes of sagittal and tangential 
magnifications. 




Fig. 9L. Graphs comparing coma with longitudinal spherical aberration for a series of 
lenses having different shapes. 



LENS ABERRATIONS 



145 



The shape of the comatic figure is given by 

y = C.(2 + cos 2fi z = C. sin 2^ 

which shows that the tangential coma C t is three times the sagittal coma 
C. [see Fig. 9K(b)]. Thus 

C t = 3C. 

To see how coma is affected by changing the shape of a lens a graph 
of the height of the comatic figure, C t , is plotted against the shape factor 
q in Fig. 9L. The numerical values plotted in this graph are calculated 
from Eq. 9m and listed in Table 9-IV. 

Table 9-1 V. Comparison of Coma and Spherical Aberration for Lenses 

of the Same Focal Length but Different Shape Factor 

h = 1.0 cm, / = + 10.0 cm, y = 2.0 cm, n = 1.5000 



Shape of lens 


Shape 
factor 


Coma 


Spherical 
aberration 


1. Concavo-convex 

2. Plano-convex 


-2.0 
-1.0 
-0.fi 


+0.5 
+ 1.0 
+2.0 


-0.0420 cm 

-0.0270 

-0.0195 

-0.0120 

-0.0045 

+0.0030 

+0.0180 


+0.88 cm 

+0.43 

+0.26 




+0.15 




+0.10 




+0.11 


7. Concavo-convex 


+0.29 



A parallel beam of light is assumed to be incident on the lens at an 
angle of 11° with the axis. The values of the longitudinal spherical 
aberration, given for comparison purposes, are also calculated from third- 
order theory (Eq. 9e) and assume parallel light incident on the lens 
parallel to the axis and passing through the same zone. 

The fact that the line representing coma crosses the zero axis indicates 
that a single lens can be made that is entirely free of this aberration. 
It is important to note, for the lenses shown, that the shape factor 
q = 0.800 for no coma is so near the shape factor q = 0.714 for minimum 
spherical aberration that a single lens designed for Ct = will have 
practically the minimum amount of spherical aberration. 

In order to calculate the value of q that will make Eq. 9m vanish, C, is 
set equal to zero. There results 

G 



Q= ~ W P 



(9n) 



If the shape and position factors of a single lens obey this relation, the 
lens is coma-free. A doublet designed to correct for spherical aberration 



146 



GEOMETRICAL OPTICS 



Margin 



can at the same time be corrected for coma. A graph showing the resid- 
ual spherical aberration and coma for a telescope objective is given in 
Fig. 9M. 

9.8. Aplanatic Points of a Spherical Surface. An optical system free 
of both spherical aberration and coma is said to be aplanatic. The 
significance of an aplanatic surface in the simple case of a single surface 
has already been discussed in Sec. 1.6. An aplanatic lens may also be 

found for any particular pair of con- 
jugate points, although in general it 
will need to be an aspherical lens. Ex- 
cept for a few special cases, no lens com- 
bination with spherical surfaces is com- 
pletely free of both these aberrations. 

One special case which is of consider- 
able importance in microscopy is that 
of a single spherical refracting surface. 
To demonstrate the existence of apla- 
natic points for a single surface, a useful 
construction, originally discovered by 
Huygens, will first be described. In 
Fig. 9iV(a) the ray R T represents any 
ray in the first medium, of index n, in- 
cident on the surface at T and making 
an angle <£ with the normal NC. Around 
C as a center and with radii 




t 



9.90 9.95 10.0 10.05 

Fig. 9Af . Curves for a cemented dou- 
blet, showing the variable position 
of the focal point F' (longitudinal 
spherical aberration) and the vari- 
able focal length /' (coma = H'F' 
-/')■ 



n 



and 



p' - r 



(9o) 



the broken circular arcs are drawn as 
shown. Where RT, when produced, 
intersects the larger circle, a line JC is drawn, and this intersects the 
smaller circle at K. Then TK gives the direction of the refracted ray in 
accordance with the law of refraction.* Furthermore any ray whatever 
directed toward J will be refracted through K. 

The aplanatic points of a single surface are located where the two 
construction circles cross the axis [see Fig. 9JV(6)]. All rays initially 
traveling toward M will pass through M' , and similarly all rays diverging 
from M' will after refraction appear to originate at M. The application 
of this principle to a microscope is illustrated in Fig. 90. A drop of oil 
having the same index as the hemispherical lens is placed on the micro- 
scope slide and the lens lowered into contact as shown. All rays from an 

* For a proof of this proposition, see J. P. C. Southall, "Mirrors, Prisms, and 
Lenses," 3d ed., p. 512, The Macmillan Company, New York, 1936. 



LENS ABERRATIONS 



147 




[a) 



ib) 



Fig. 9N. (a) A graphical construction for refraction at a single spherical surface. 
p = rn'/n, and p' = rn/n'. (b) Location of the aplanatic points of a single spherical 
surface. 



object at M leave the hemispherical sur- 
face after refraction as though they came 
from M' , and this introduces a lateral 
magnification of M' A/MA. If a second 
lens is added which has the center of its 
concave surface at M' (and therefore is 
normal to all rays) , refraction at its upper 
surface, of radius n' X CM', will give 
added magnification without introducing 
spherical aberration. This property of 
the upper lens, however, holds strictly 
only for rays from the single point M, 
and not for points adjacent to it. There 
is a limit to this process which is set by 
chromatic aberration (see Sec. 9.13). 

9.9. Astigmatism. If the first two 
Seidel sums vanish, all rays from points 
on or very close to the axis of a lens will 
form point images and there will be no 
spherical aberration or coma. When the 




Fig. 90. Aplanatic surfaces of the 
first elements of an oil-immersion 
microscope objective. 



148 



GEOMETRICAL OPTICS 



object point lies at some distance away from the axis, however, a point image 
will be formed only if the third sum S 3 is zero. If the lens fails to satisfy this 
third condition, it is said to be afflicted with astigmatism, and the resulting 
blurred images are said to be astigmatic. The formation of real astig- 
matic images from a concave spherical mirror is discussed in Sec. 6.9. 
To help understand the formation of astigmatic images by a lens, a ray 



Tangential 




Fig. 9P. (a) Perspective diagram showing the two focal lines which constitute the 
image of an off -axis object point Q. (b) Loci of the tangential and sagittal images. 
The two surfaces approximate paraboloids of revolution. 

diagram has been drawn in perspective in Fig. 9F(a). Considering the 
rays from a point object Q, all those in the fan contained in the vertical or 
tangential plane cross at T, while the fan of rays in the horizontal or 
sagittal plane crosses at S. The tangential and sagittal planes intersect 
the lens in RS and JK, respectively. Rays in these planes are chosen 
because they locate the two focal lines T and S formed by all rays going 
through the lens. These are perpendicular to their respective tangential 
and sagittal planes. At L the image is approximately disk-shaped, and 
constitutes the circle of least confusion for this case. 

If the positions of the T and S images are determined for a wide field 
of distant object points, their loci will form paraboloidal surfaces whose 
sections are shown in Fig. 9/^(6). The amount of astigmatism, or astig- 



LENS ABERRATION'S 



149 



ma tic difference, for any pencil of rays is given by the distance between 
these two surfaces measured along the chief ray. On the axis, where the 
two surfaces come together, the astigmatic difference is zero; away from 
the axis it increases approximately as the square of the image height. 
Astigmatism is said to be positive when the T surface lies to the left 
of S, as shown in the diagram. It should be noted that for a concave 
mirror (Fig. 60), the sagittal surface is a plane coinciding with the paraxial 
focal plane. 

If, as in Fig. 9Q, the object is a spoked wheel in a plane perpendicular 
to the axis with its center at M, 
the rim would be found to be in 
focus on the T surface while the 
spokes would be in focus on the S 
surface. It is for this reason 
that the terms "tangential" and 
' ' sagittal ' ' are applied to the planes 
and images. On the surface T all 
images will be lines parallel to the 
rim as shown at the left in Fig. 
9Q, while on the surface S all 
images will be lines parallel to the spokes as shown at the right. 

Equations giving the astigmatic image distances for a single refracting 
surface are* 




Fig. 9Q. Astigmatic images of a spoked 
wheel. 



n cos 2 <f> n' cos 2 <f>' 

Z i 17 



-> 



n' cos <j>' — n cos <j> 

r 
n' cos <$>' — n cos 



(9p) 



where <f> and <f>' are the angles of incidence and refraction of the chief ray, 
r the radius of curvature, s the object distance, and s, and s„ the T and S 
image distances, the latter being measured along the chief ray. For a 
spherical mirror these equations reduce to 



1 



1 



/ cos 4> 



and 



s s. 



cos <t> 



Coddington has shown that for a thin lens in air with an aperture stop 
at the lens, the positions of the tangential and sagittal images are given by 



!+-U 

s s, 



1 



1 i 1 

— — , = cos 

s s, 



_ ( n cos </>' _ A/1 _ l\ 
<t>\ cos <t> J \n r 2/ 

/. cos ♦' _ ,\A _ A 

\ cos 4> / Vi r 2 / 



Off) 



* For a derivation of these formulas see G. S. Monk, "Light, Principles and Experi- 
ments," 1st ed., p. 424, McGraw-Hill Book Company, Inc., New York, 1937. 



150 



GEOMETRICAL OPTICS 



The angle <f> is the angle of obliquity of the incident chief rays, and <p' 
the angle of this ray within the lens. Therefore n = sin 0/sin <j>'. The 
application of these formulas to thin lenses shows that the astigmatism is 
approximately proportional to the focal length and is very little improved 
by changing the shape. 

Although a contact doublet composed of one positive and one negative 
lens shows considerable astigmatism, the introduction of another element 
consisting of a stop or a lens can be made to greatly reduce it. By the 
proper spacing of the lens elements of any optical system, or by the 
proper location of a stop if one is used, the curvature of the astigmatic 




P S B T 


\ \ 


1 


\\ 


\ 


\ 


1 


\ 





F' 

(a) (6) (c) (d) 

Fig. QR. Diagrams showing the astigmatic surfaces T and S in relation to the fixed 
Petzval surface P, as the spacing between lenses (or between lens and stop) is changed. 

image surfaces can be changed considerably. Four important stages in 
the flattening of the astigmatic surfaces due to these alterations are 
shown in Fig. 9R. Diagram (a) represents the normal shape of the T 
and S surfaces for a contact doublet or a single lens. In diagram (b) the 
separation of lens elements is such that the two surfaces fall together at 
P. Further alteration of the lens shapes and their spacing may be made 
and the T and S curves straightened, as in diagram (c), or moved still 
farther apart until they are bisected by the normal plane through the 
focal point F', as in diagram (d). Of these four arrangements, only the 
second is free of astigmatism. The single paraboloidal surface P, over 
which point images are formed, is called the Petzval surface. 

9.10. Curvature of Field. If for an optical system the first three 
Seidel sums are zero, the system will form point images of point objects 
on as well as off the axis. Under these circumstances the images fall on 
the curved Petzval surface where the tangential and sagittal surfaces 
come together, as in Fig. 9R(b). Even though astigmatism is corrected 
for such a system, the focal surface is curved. If a flat screen is placed 
in position B, the center of the field will be in sharp focus but the edges 
will be quite blurred. With a screen at A, the center of the field and 
the field margins will be blurred, while sharp focus will be obtained about 
halfway out. 



LENS ABERRATIONS 



151 



Mathematically a Petzval surface exists for every optical system, and 
if the powers and refractive indices of the lenses remain fixed the shape 
of the Petzval surface cannot be changed by altering the shape factors 
of the lenses or their spacing. Such alterations, however, will change 
the shapes of the T and S surfaces, but always in such a way that the 
ratio of the distances PT and PS is 3 : 1. It will be noted that this ratio 
is maintained throughout Fig. 9R. If a system is designed to make 
the T surface flat, as in Fig. 972(c), the 3:1 ratio of distances requires 
the S surface to be curved, but not strongly so. If a screen is placed 



Stop 



S A T 



Axis 



A 




F' 
W (ft) 

Fig. 95. (a) A properly located stop may be used to reduce field curvature. (6) Astig- 
matic surfaces for an "antistigmat" camera lens. 

at a compromise position A, the images over the entire field will be in 
reasonably good focus. This condition of correction is commonly used 
for certain types of photographic lenses. If more negative astigmatism 
is introduced the condition shown in Fig. 9R(d) is reached, in which the 
T surface is convex and the S surface is concave by an equal amount. 
In this case a screen placed at the paraxial focus will show considerable 
blurring at the field edges. 

Curvature of field may be corrected for a single lens by means of a 
stop. Acting as a second element of the system, a stop limits the rays 
from each object point in such a way that the paths of the chief rays 
from different points go through different parts of the lens [Fig. 9S(a)]. 
Certain manufacturers of inexpensive box cameras employ a single menis- 
cus lens and a stop and with them obtain reasonably good images. The 
stop is located in front of the lens, with the light incident on the concave 
surface. Although the compromise field is flat and sharp focus is 
obtained at the center, astigmatism gives rise to blurred images at the 
margins. 

In complex lens systems it is possible, because of differences in third- 
and fifth-order corrections, to control the astigmatism and cause the 
tangential and sagittal surfaces to come together at an outer zone as well 
as at the center of the field. Typical curves for the camera objective 



152 



GEOMETRICAL OPTICS 



called an " anastigmat " are shown in Fig. 9*S'(6). Experience has shown 
that the best state of correction is obtained by making the crossover 
point, called the node, occur at a relatively short distance in front of the 
focal plane. 

9.11. Distortion. Even though an optical system were designed so that 
the first four Seidel sums were zero, it could still be affected by the fifth 
aberration known as distortion. To be free of distortion a system must 




Axis 



(6) (c) U) 

Fig. 97\ (a) A pinhole camera shows no distortion. Images of a rectangular object 
screen shown with (b) no distortion, (c) barrel distortion, and (d) pincushion distortion. 



have uniform lateral magnification over its entire field. A pinhole cam- 
era is ideal in this respect for it shows no distortion; all straight lines 
connecting each pair of conjugate points in the object and image planes 
pass through the opening. Constant magnification for a pinhole camera 
as well as for a lens implies, as may be seen from Fig. 97'(a), that 



tan </>' 
tan 



= const. 



The common forms of image distortion produced by lenses are illustrated 
in the lower part of Fig. 9T. Diagram (b) represents the undistorted 
image of an object consisting of a rectangular wire mesh. The second 
diagram shows barrel distortion, which arises when the magnification 
decreases towards the edge of the field. The third diagram represents 



LENS ABERRATIONS 



153 



pincushion distortion, corresponding to a greater magnification at the 

borders. 

A single thin lens is practically free of distortion for all object distances. 
It cannot, however, be free of all the other aberrations at the same time. 



(a) 





Stop Jfc 


t 


Chief ray ' — __ 


Q'z 


M 


"«— — .. . \ 




\^ 


/ 


Axis ______ — -—-" 


M 



ib) 




Id 



Fig. 9 U. (a) A stop in front of a lens giving rise to barrel distortion. (6) A stop behind 
a lens giving rise to pincushion distortion, (c) A symmetrical doublet with a stop be- 
tween is relatively free of distortion. 

If a stop is placed in front of or behind a thin lens, distortion is invari- 
ably introduced; if it is placed at the lens, there is no distortion. Fre- 
quently in the design of good camera lenses astigmatism, as well as 
distortion, is corrected for by a nearly symmetrical arrangement of two 
lens elements with a stop between them. 

To illustrate the principles involved, consider the lens shown in Fig. 
9U(a), which has a front stop. Rays from object points like M, at or 
near the axis, go through the central part of the lens, while rays from 
off-axis object points like Q 2 are refracted only by the upper half. In 
the latter case the stop decreases the ratio of image to object distances 



154 GEOMETRICAL OPTICS 

measured along the chief ray, thereby reducing the lateral magnification 
below that obtaining for object points near the axis. This system there- 
fore suffers from barrel distortion. When the lens and stop are turned 
around, as in Fig. 9 U(b), the ratio of image to object distances is seen to 
increase as the object point lies farther off the axis. The result is 
increased magnification and pincushion distortion. 

By combining two identical lenses with a stop midway between them 
as in Fig. 9C7(c), a system is obtained which because of its symmetry is 
free from distortion for unit magnification. With other magnifications 
however the lenses must be corrected for spherical aberration with 
respect to the entrance and exit pupils. These two pupils S' and S" 
coincide with the principal planes of the combination. Such a corrected 
lens system is called an orthoscopic doublet, or rapid rectilinear lens. 
Because this combination cannot be corrected for spherical aberration 
for the object and image planes and for the entrance and exit pupils at 
the same time, the lens suffers from this aberration as well as from 
astigmatism. Photographic lenses of this type are discussed in Sec. 10.4. 

Summarizing very briefly the various methods of correcting for aberra- 
tions, spherical aberration and coma can be corrected by using a contact 
doublet of the proper shape; astigmatism and curvature of field require 
for their correction the use of several separated components; and distor- 
tion may be minimized by the proper placement of a stop. 

9.12. The Sine Theorem and Abbe's Sine Condition. In Chap. 3 it 
was found that the lateral magnification produced by a single spherical 
surface is given by the relation (Eq. 3o) 

n,-*' - S ' " r 

m = — = : — 

y 8 -t- r 

This equation follows from the similarity of triangles MQC and M'Q'C 
in Fig. 3F. 

From Eq. 8a we obtain the exact relation 

sin d> 

s + r = r- — - 

sin 

and from Eq. 8d 

sin <b' 

s' — r = —r -. — -rf 

sin 0' 

If we substitute these two equations in the first equation, we obtain 

y' sin </>' sin 



According to Snell's law 



y sin 0' sin <f> 

sin </>' _ n_ 
sin </> n' 



LENS ABERRATIONS 



155 



which upon substitution gives 

y' n sin 6 



or 



y n' sin 6' 
ny sin 6 = n'y' sin 6' 



SINE THEOREM 



Here y and ?/' are the object and image heights, n and n' are the indices 
of the object and image spaces, and and 6' are the slope angles of the 
ray in these two spaces, respectively (see Fig. 91*0 . This very general 
theorem applies to all rays, no matter how large the angles and 6' may 
be. 




Fig. 9V. Refraction at a spherical surface illustrating the sine theorem as it applies to 
coma. 

For paraxial rays where 6 and 6' are both small, sin and sin 6' can 
be replaced by 6 P and d' p , respectively, to give 

nydp = n'y'd' p LAGRANGE theorem 

a relation referred to as the Lagrange theorem. In both these theorems 
all quantities on the left side refer to object space, while those on the 
right side refer to image space. 

Figure 9V shows a pair of sagittal rays QR and QS from the object 
point Q through one zone of a single refracting surface. These two 
particular rays, after refraction, come to a focus at a point Q' s on the 
auxiliary axis. On the other hand, a pair of tangential rays QT and 
QU through the same zone come to a focus at Q' T , while paraxial rays 
come to a focus at Q'. Because of the general spherical aberration and 
astigmatism of the single surface the paraxial, the sagittal, and the 
tangential focal planes do not coincide. The conventional comatic 
figure shown at the right in Fig. 97 arises only in the absence of spherical 
aberration and astigmatism. 

Since coma is confined to lateral displacements in the image in which 
y and y' are relatively small, we can neglect astigmatism and apply 
the above theorems to the single surface as follows: Note that and 6' 
for the object point Q, which are the slope angles of the zonal rays QS 



156 GEOMETRICAL OPTICS 

and Q'.S relative to the chief ray (c.r.), are virtually equal to the slope 
angles of the rays from the axial object point M through the same zone 
of the surface. We can, therefore, apply the sine theorem to find the 
sagittal image magnification for any zone and obtain 

^ = rf = » sin 



y n' sin 0' 

where y' t = Q' t M'„ in Fig. 9V. 

To show that the sine theorem and the Lagrange theorem can be 
extended to a complete optical system containing two or more lens 
surfaces, we recognize that in the image space of the first lens surface 
the two products are n[y[ sin B\ and n[y[0' pl , respectively. These prod- 
ucts are identical for the object space of the second surface because 
n '\ = n 2, y'\ — V2, and 0[ = 2 ; hence the products are invariant for all 
the spaces in the system including the original object space and the final 
image space. This is a most important property. 

Now for a complete system to be free of coma and spherical aberration 
it must satisfy a relation known as the sine condition. This is a condition 
discovered by Abbe, in which the magnification for each zone of the 
system is the same as for paraxial rays. In other words, if in the final 
image space y' t = y', and m. = m, we may combine the two preceding 
equations and obtain 

sin d p 

zr, = -T7 = COnst. SINE CONDITION (9r) 

sin 6 P v ' 

Any optical system is therefore free of coma, if in the absence of 

spherical aberration sin 0/sin 6' = 
const, for all values of 6. In 
lens design coma is sometimes 
tested for by plotting the ratio 
sin 0/sin 0' against the height of 
the incident ray. Because most 
lenses are used with parallel in- 
Fig. 9W. For a lens to be free of spherical cident or emergent light, it is cus- 
aberration and coma the principal surface tomary to replace sin by h, the 
should be spherical and of radius /'. • .j.. » >■» , ,, 

J height of the ray above the axis, 

and to write the sine condition in the special form 

ihTT' = COnst - < 9s > 

The ray diagram in Fig. 9TT shows that the constant in this equation 
is the focal distance measured along the image ray, which we here call/'. 
To prevent coma, /' must be the same for all values of h. Since freedom 




LENS ABERRATIONS 



157 



from spherical aberration requires that all rays cross the axis at F', an 
accompanying freedom from coma requires that the principal "plane" 
be a spherical surface (represented by the dotted line in the figure) of 
radius/'. It is thus seen that, whereas spherical aberration is concerned 
with the crossing of the rays at the focal point, coma is concerned with 
the shape of the principal surface. It should be noted that the aplanatic 
points of a single spherical surface (see Sec. 9.8) are unique in that they are 



White 




Lateral 

chromatic 

aberration 




longitudinal 
chromatic 
aberration 



Image 
plane 



Fig. 9X. (a) Chromatic aberration of a single lens. (6) A cemented doublet corrected 
for chromatic aberration, (c) Illustrating the difference between longitudinal chro- 
matic aberration and lateral chromatic aberration. 

entirely free of spherical aberration and coma and satisfy the sine condi- 
tion exactly. 

9.13. Chromatic Aberration. In the discussion of the third-order 
theory given in the preceding sections, no account has been taken of the 
change of refractive index with color. The assumption that n is con- 
stant amounts to investigating the behavior of the lens for monochro- 
matic light only. Because the refractive index of all transparent media 
varies with color, a single lens forms not only one image of an object 
but a series of images, one for each color of light present in the beam. 
Such a series of colored images of an infinitely distant object point on 
the axis of the lens is represented diagramatically in Fig. 9X(a). The 
prismatic action of the lens, which increases toward its edge, is such as 
to cause dispersion and to bring the violet light to a focus nearest to 
the lens. 



158 



GEOMETRICAL OPTICS 



As a consequence of the variation of focal length of a lens with color, 
the lateral magnification must vary as well. This may be seen by the 
diagram of Fig. 9A(c), which shows only the red and violet image heights 
of an off -axis object point Q. The horizontal distance between the axial 
images is called axial or longitudinal chromatic aberration, while the 
vertical difference in height is called lateral chromatic aberration. Because 
these aberrations are often comparable in magnitude with the Seidel 



1.56 



1.55 - 



1.54- 



1.53 - 



1.52 - 



1.51 - 



1.50 




Violet Blue Green Yellow • Red 

Fig. 9F. Graphs of the refractive indices of several kinds of optical glass. These are 
called dispersion curves. 

aberrations, correction for both lateral and longitudinal color is of con- 
siderable importance. As an indication of relative magnitudes, it may 
be noted that the longitudinal chromatic aberration of an equiconvex lens 
of spectacle crown glass having a focal length of 10 cm and a diameter of 
3 cm is exactly the same (2.5 mm) as the spherical aberration of marginal 
rays in the same lens. 

While there are several general methods for correcting chromatic 
aberration, the method of employing two thin lenses in contact, one 
made of crown glass and the other of flint glass, is the commonest and 
will be considered first. The usual form of such an achromatic doublet 
is shown in Fig. 9A"(6). The crown-glass lens, which has a large positive 
power, has the same dispersion as the flint-glass lens, for which the power 
is smaller and negative. The combined power is therefore positive, while 
the dispersion is neutralized, thereby bringing all colors to approximately 
the same focus. The possibility of achromatizing such a combination 



LENS ABERRATIONS 



159 



rests upon the fact that the dispersions produced by different kinds of 
glass are not proportional to the deviations they produce (Sec. 1.7). In 
other words, the dispersive powers 1/v differ for different materials. 

Typical dispersion curves showing the variation of n with color are 
plotted for a number of common optical glasses in Fig. 9F, and the actual 
values of the index n for the different Fraunhofer lines are presented in 
Table 9-V. The peak of the visual brightness curve* in Fig. 9F occurs 

Table 9-V. Refractive Indices of Typical Optical Media for Four Colors 



Medium 



Borosilicate crown 

Borosilicate crown 

Spectacle crown 

Light barium crown 

Telescope flint 

Dense barium flint 

Light flint 

Dense flint 

Dense flint 

Extra dense flint 

Fused quartz 

Crystal quartz (O ray) 
Fluorite 



Symbol 



BSC 

BSC-2 

SPC-1 

LBC-1 

TF 

DBF 

LF 

DF-2 

DF-4 

EDF-3 

SiC-2 

SiO, 

CaF 2 



I.C.T. 
type 



500/665 
517/645 
523/586 
541/599 
530/516 
670/475 
576/412 
617/366 
649/338 
720/291 



66.5 
64.5 
58.8 
59.7 
51.6 
47.5 
41.2 
36.6 
33.9 
29.1 
67.9 
70.0 
95.4 



nc 



no 



49776 1 
51462 1 



52042 
53828 
52762 
66650 
57208 
61216 
64357 
71303 



50000 

51700 

52300 

54100 

53050 

67050 

57600 

61700 

64900 

72000 

4585 

5443 

4338 



n P 



.50529 
. 52264 
.52933 
.54735 
.53790 
.68059 
. 58606 
.62901 
.66270 
. 73780 



nr/ 



. 50937 
.52708 
.53435 
.55249 
.54379 
.68882 
.59441 
.63923 
.67456 
.75324 



not far from the yellow D line. It is for this reason that the index n D 
has been chosen by optical designers as the basic index for ray tracing 
and for the specification of focal lengths. Two other indices, one on 
either side of n D , are then chosen for purposes of achromatization. As 
indicated in the table, the ones most often used are nc for the red end 
of the spectrum and n F or nc for the blue end. 

For two thin lenses in contact, the resultant focal length /z> or power 
Pd of the combination for the D line is given by Eqs. 4/i and 4&: 



1 = 4+4 

Id f D Sd 



or 



Pd = P' D + P£ 



(9/) 



where the index D indicates that the quantity depends on n D , f' D and P' D 
refer to the focal length and power of the crown-glass component, and 
fo and P^ to the focal length and power of the flint-glass component. 

* Brightness is a sensory magnitude in light just as loudness is a sensory magnitude 
in sound. Over a considerable range both vary approximately as the logarithm of the 
energy. The curve shown represents the logarithms of the "standard luminosity 
curve." 



160 GEOMETRICAL OPTICS 

In terms of indices of refraction and radii of curvature, the power form 
of the equation becomes 



*-04-i>d-£j + (*-i>(^-£) 



(9«) 



For convenience let 

K ' = (k~$ and K " = (k-*) (9w ' } 

Then Eq. (9w) can be more simply written as 

Pd = (ri D - \)K' + (nJJ - \)K" (9i>) 

Similarly, for any other colors or wavelengths like the F and C spectrum 
lines, we may write 

P F = (n' F - l)K' + W ~ IW I (w 

Pc = (n' c - l)K' + « - \)K" ) 

To make the combination achromatic we make the resultant focal length 
the same for F and C light. This means, making P F = P c , 

(n' F - \)K' + W - \)K" m {n' G - \)K' + « - \)K" 

Multiplying out and canceling, this becomes 



K' n' F - n't 



(9v") 



K" n' F - n' c 

Since both the numerator and denominator on the right have positive 
values, the minus sign shows that one K must be negative and the other 
positive. This means that one lens must be negative. 

Now for the D line of the spectrum the separate powers of the two 
thin lenses are given by 

P' D = {n' D - l)K' and Pi = « - 1)K" (9w) 

Dividing one by the other, this gives 

K' (nJJ - l)P' D 



K" (n' D - \)n 
Equating Eqs. 9v" and 9v>' and solving for P'd/P'd gives 

11 - ("p ~ X ) ^ (^ ~ 1 ) = _ *1 

P' D " W ~ <) ' (n' F - n'c) - 

where v' and v" are the dispersion constants of the two glasses. 



(9w') 



(9i0") 



LENS ABERRATIONS 



161 



These constants, usually supplied by manufacturers when optical glass 
is purchased, are, 






and 



n'£- 1 

n F — n c 



(9x) 



Values of v for several common types of glass are given in Table 9-V. 
Since the dispersive powers are all positive, the negative sign in Eq. 9w" 
indicates that the powers of the two lenses must be of opposite sign. In 
other words, if one lens is converging the other must be diverging. From 
the extreme members of Eq. 9w", we obtain 



P' P" 

v' ^ v" U 



or 



//' + „"/" = o 



(9*') 



Substituting the value of P' D or that of P'£ from Eq. 9/ in Eq. 9x', we 
obtain 



r. - P* (y4-r) 



and 



PZ = -P 



D \v^v) 



(9x") 



The use of the above formulas to calculate the radii for a desired 
achromatic lens involves the following steps: 

1. A focal length fo and a power Pp are specified. 

2. The types of crown and flint glass to be used are selected. 

3. If they are not already known, the dispersion constants v' and v" 
are calculated from Eq. 9x. 

4. P' D and P'£ are calculated from Eq. 9x". 

5. The values of K' and K" are determined by Eq. 9w. 

6. The radii are then found from Eq. 9m'. 

Calculation 6 is usually made with other aberrations in mind. 

Example: An achromatic lens having a focal length of 10 cm is to be 
made as a cemented doublet using crown and flint glasses having the 
following indices: 



Glass 


nc 


riD 


np 


no' 


1. Crown 

2. Flint 


1.50868 
1.61611 


1.51100 
1.62100 


1.51673 
1.63327 


1.52121 
1 64369 







Find the radii of curvature for both lenses if the crown-glass lens is to be 
equiconvex and the combination is to be corrected for the C and F lines. 



162 GEOMETRICAL OPTIC'S 

Solution: The focal length of 10 cm is equivalent to a power of +10 D. 
The dispersion constants v' and v" are, from Eq. 9x, 

1.51100 - 1.00000 _ ^^ 



1.51673 - 1.50868 
1.62100 - 1.00000 



- 36.1888 



1.63327 - 1.61611 
Applying Eq. 9x", we find that the powers of the two lenses must be 

P ° = 10 63.4783 3 - 8 36.1888 =+23 - 26nD 

^ -~ 10 68.47^ -H-lSSg -- 13 - 26 "" 

The fact that the sum of these two powers P/> = +10.0000 D serves as 
a check on the calculations to this point. Knowing the power required 
in each lens, we are now free to choose any pair of radii that will give 
such a power. If two or more surfaces can be made to have the same 
radius, the necessary number of grinding and polishing tools will be 
reduced. For this reason the positive element is often made equiconvex, 
as it is here. Letting r[ = —r' 2> we apply Eq. 9u' and then Eq. 9w to 
obtain 

* 1 1 » *L . «g"g_ 4UB07 

r\ r 2 r[ n D — 1 0.51100 

from which r[ = 0.0439361 m = 4.39361 cm 

Since the lens is to be cemented, one surface of the negative lens must fit 
a surface of the positive lens. This leaves the radius of the last surface 
to be adjusted to give the proper power of —13.2611 D. Therefore we 
let r'i = —r[, and apply Eqs. 9u' and 9io as before, to find 

K" = 1 - 1 = - l - 1 - n . ~ 13 - 2611 = -21 3544 

r[' rj 0.0439361 rj nJJ - 1 0.62100 



This gives 






4r = 21.3544 - 




22.7603 


0.0439361 " 1 * dD11 


and -77 = -1.4059 


r" = -0.71129 m = 


-71.13 cm 



The required radii are therefore 

r'j = 4.39 cm r[' = -4.39 cm 

r' 2 = -4.39 cm r'i = -71.13 cm 

It will be noted that, with the crown-glass element of this achromat placed 
toward incident parallel light, the two exposed surfaces are close to what 
they should be for minimum spherical aberration and coma. This 



LENS ABERRATIONS 163 

emphasizes the importance of choosing glasses having the proper dis- 
persive powers. 

To see how well this lens has been achromatized, we now calculate its 
focal length for the four colors corresponding to the C, D, F, and G' 
lines. By Eq r 9»' 

Pc = (n' c - 1)K' + « - \)K" 

= 0.50868 X 45.5207 + 0.61 611 (-21. 3544) 
= 23.1555 - 13.1567 
giving f c = 10.0012 cm 

Similarly for the colors corresponding to the F and G' lines we obtain 

Pf = +9.9988 D or f F = 10.0012 cm 
Pc = +9.9804 D or / G < = 10.0196 cm 

The differences between f c , f D , and f F are negligibly small, but /<?' is 
about | mm larger than the others. This difference for light outside 
the region of the C and F lines results in a small circular zone of color 
about each image point which is called the secondary spectrum. 

Although the lens in our example would appear to have been corrected 
for longitudinal chromatic aberration, it has actually been corrected for 
lateral chromatic aberration. Equal focal lengths for different colors will 
produce equal magnification, but the different colored images along the 
axis will coincide only if the principal points also coincide. Practically 
speaking, the principal points of a thin lens are so close together that 
both types of chromatic aberration can be assumed to have been cor- 
rected by the above arrangement. In a thick lens, however, longitudinal 
chromatic aberration is absent if the colors corrected for come together 
at the same axial image point as shown in Fig. 9Z(a). Because the prin- 
cipal points for blue and red H' h and H' r do not coincide, the focal lengths 
are not equal and the magnification is different for different colors. Con- 
sequently the images formed in different colors will have different sizes. 
This is the lateral chromatic aberration or lateral color mentioned at the 
beginning of this section. 

9.14. Separated Doublet. Another method of obtaining an achromatic 
system is to employ two thin lenses made of the same glass and separated 
by a distance equal to half the sum of their focal lengths. To see why 
this is true, we begin with the thick-lens formula (Eq. 5g) as applied to 
two thin lenses separated by a distance d: 

f = h + h ~ fji ° r P = Pl + Pa ~ dPlI>2 (9y) 

which, by analogy with Eq. 9v, may be written 

P = («i - i)Xi + (n 8 - 1)K 2 - d(m - l)(n 2 - l)KiK a 



164 



GEOMETRICAL OPTICS 



[«] 




m 









i^^^* 5 


MBki&s 








H 


^Red 




Hi 


K 1 






Axis 



Red image 



Images 
Blue Red 



lei 




Fig. 9Z. (o) Cemented doublet corrected for longitudinal chromatic aberration. 
(6) Separated doublet corrected for longitudinal chromatic aberration, (c) Separated 
doublet corrected for lateral chromatic aberration. 



The subscripts 1 and 2 are used here in place of the primes to designate 
the two lenses, and the K's are given by Eq. 9w'. Since the two lenses 
are of the same kind of glass, we set n x = w 2 , so that 

p = {n - 1)(K 1 + K 2 ) - d(n - l)*KiKa 

If this power is to be independent of the variation of n with color, dP/dn 
must vanish. This gives 

j- = K x + Ko - 2d(n - l)K,K 2 = 



Multiplying by n — 1 and substituting for each (n — Y)K the corre- 



LENS ABERRATIONS 165 

sponding P, we find 

Pi + F 2 - 2dP x Pi = 

d -wr and '-^t^ 1 w 

This proves the proposition stated above that two lenses made of the 
same glass separated by half the sum of their focal lengths have the same 
focal length for all colors near those for which /i and / 2 are calculated. 
For visual instruments this color is chosen to be at the peak of the visual 
brightness curve (Fig. 9F). Such spaced doublets are used as oculars 
in many optical instruments because the lateral chromatic aberration is 
highly corrected through constancy of the focal length. The longitudinal 
color, however, is relatively large, due to wide differences in the principal 
points for different colors. An illustration of a system that has no 
longitudinal chromatic aberration is shown in Fig. 9Z(o). It is to be 
contrasted with the system shown in Fig. 9Z(c), in which there is no 
lateral chromatic aberration. 

We have seen in this chapter that a lens may be affected by as many 
as seven primary aberrations — five monochromatic aberrations of the 
third and higher orders, and two chromatic aberrations. One might 
therefore wonder how it is possible to make a good lens at all when rarely 
can a single aberration be eliminated completely, much less all of them 
simultaneously. Good usable lenses are nevertheless made by the proper 
balancing of the various aberrations. The design is guided by the pur- 
pose for which the lens is to be used. In a telescope objective, for 
example, correction for chromatic aberration, spherical aberration, and 
coma are of primary importance. On the other hand astigmatism, curva- 
ture of field, and distortion are not as serious because the field over which 
the objective is to be used is relatively small. For a good camera lens 
of wide aperture and field, the situation is almost exactly reversed. 

Other treatments of the subject of aberrations will be found in the 
following texts: 

"The Principles of Optics," by A. C. Hardy and F. H. Perrin. 
"Light, Principles and Experiments," by G. S. Monk. 
"Fundamentals of Optical Engineering," by D. H. Jacobs. 
"Applied Optics and Optical Design," by A. E. Conrady. 
"Technical Optics," by L. C. Martin. 

"A Treatise on Reflexion and Refraction," by H. Coddington. 
"A System of Applied Optics," by H. D. Taylor. 

PROBLEMS 

1. A single spherical surface of radius r = +10 cm separates two media of index 
n = 1.2 and n' = 1.5, respectively. Calculate (a) the longitudinal and (6) the lateral 
spherical aberration for parallel incident light through a zone at height h = 1.0 cm. 



166 GEOMETRICAL OPTICS 

2. A convex surface of 4 cm radius is polished on the end of a glass rod of index 1.60. 
Calculate (a) the longitudinal and (b) the lateral spherical aberration for parallel 
incident light through a zone at height h = 1 cm. Ans. (a) 1.29 mm. (6) 0.12 mm. 

3. A thin lens of index 1.5 has radii n = +60 cm and r a = —12 cm. If the lens 
is used with parallel incident light, find (a) the longitudinal and (6) the lateral spherical 
aberration for rays through a zone at height h = 2 cm. 

4. A thin lens of index 1.50 has radii n = +10 cm and r 2 = —10 cm. Find (a) 
the longitudinal and (6) the lateral spherical aberration for an axial object point 
20 cm in front of the lens and for rays through a zone of radius h = 1 cm. 

Ans. (a) 4.400 mm. (b) 0.225 mm. 

5. A thin lens of index 1.60 has radii n = +10 cm and r 2 = —10 cm. Find (a) 
the longitudinal and (6) the lateral spherical aberration for an axial object point 
24 cm in front of the lens and for rays through a zone of radius h = 1 cm. 

6. A thin lens of index 1.60 has radii n = +36 cm and r 2 = —18 cm. If this 
lens is to be used with parallel incident light, find (a) the longitudinal and (b) the 
lateral spherical aberration for rays through a zone of radius h = 1 cm. 

Ans. (a) 0.974 mm. (b) 0.049 mm. 

7. A thin lens of index 1.50 has radii ri = —12 cm and r 2 = +60 cm. If this 
lens is to be used with parallel incident light, find (a) the longitudinal and (b) the 
lateral spherical aberration for rays through a zone of radius h = 2 cm. 

8. A thin lens of index 1.60 has radii r t = —36 cm and r 2 = +18 cm. Find (a) 
the longitudinal and (6) the lateral spherical aberration for parallel incident light 
through a zone of radius h = 1 cm. Ans. (a) —0.974 mm. (6) 0.049 mm. 

9. A lens 6 cm in diameter and of index 1.50 has radii n = "> and r 2 = —10 cm. 
Find the height of the comatic figure if the paraxial image point of parallel incident 
rays is 4 cm off the principal axis. 

10. A thin lens 4 cm in diameter and of index 1.50 has radii r t = +20 cm and 
r 2 = —20 cm. Find the height of the comatic figure if the paraxial image point of 
parallel incident rays is 4 cm off the principal axis. Ans. —0.120 mm. 

11. A thin lens is to be made of glass of index n = 1.65 and is to have a minimum 
spherical aberration when the object is 20 cm in front of the lens and the real image 
is 80 cm in back of the lens. Determine (a) the position factor, (6) the shape factor, 
(c) the focal length of the lens, and (d) the radii of curvature. 

12. A thin lens is to be made of flint glass of index 1.75 and is to have a focal length 
of +5 cm. An object is located 30 cm in front of the lens. Determine (a) the image 
distance, and (6) the position factor. If this lens is to have a minimum spherical 
aberration for these object and image distances, find (c) the shape factor, and (d) 
the radii of curvature of the two faces. 

Ans. (a) +6.0 cm. (6) -0.667. (c) +0.733. (d) +4.33 cm, -28.12 cm. 

13. A thin lens is to be made of glass of index 1.50 and is to have a minimum lateral 
spherical aberration for distant objects. If the focal length is to be +5 cm, find (a) 
the position factor, (b) the shape factor, and (c) the radii of curvature of the two faces. 

14. A thin lens is to be made of glass of index 1.550 and is to have a focal length of 
+20 cm. Find (a) the position factor, (6) the shape factor, and (c) the radii of curva- 
ture of the two faces if it is to have a minimum amount of spherical aberration for 
an object placed at its first focal point. 

Ans. (a) +1.0. (6) -0.790. (c) +104.81 cm, -12.29 cm. 

15. A thin lens is to be made of glass of index 1.50 and is to have a focal length of 
+ 10 cm. If an object is located 12 cm in front of the lens, find (a) the image distance, 
and (6) the position factor. If the lens is to show a minimum amount of spherical 
aberration, what should be (c) its shape factor, and (d) the radii of curvature of its 
two faces? 



LENS ABERRATIONS 167 

16. Calculate (a) the shape factor, and (6) the radii of curvature of the two surfaces 
for the lens in Prpb. 1 1 if it is to have no coma. 

Ans. (a) -0.633. (6) +56.64 cm, -12.74 cm. 

17. If the lens in Prob. 12 is to have no coma, find (a) the shape factor, and (6) the 
radii of curvature of the two faces. 

18. Calculate (a) the shape factor, and (ft) the radii of curvature of the two surfaces 
for the lens in Prob. 13 if it is to have no coma. 

Ans. (a) +0.800. (ft) +2.78 cm, -25.00 cm. 

19. If the lens in Prob. 14 is to be free of coma, find (a) the shape factor, and (ft) 
the radii of curvature of the two surfaces. 

20. If the lens in Prob. 15 is to be free of coma, what should be its (a) shape factor 
and (6) its two radii of curvature? Ans. (a) -5.33. (ft) +21.43 cm, —6.52 cm. 

21. A meniscus lens 0.5 cm thick and of index 1.60 is to be aplanatic for two points 
located on the concave side of the lens. If the nearer point is to be 4 cm from the 
nearest vertex, find (a) the radii of the two lens surfaces, and (ft) the distance from the 
nearest vertex to the farther point. (Note: Both points are in air.) 

22. A meniscus lens 0.5 cm thick, and of index 1.50, is to be made aplanatic for two 
points 6 cm apart. Determine (a) the two radii of curvature, and (ft) the distances 
from the convex surface to the two points. 

Ans. (a) -11.50 cm, -7.20 cm. (6) 12.00 cm, 18.00 cm. 

23. Apply Abbe's sine condition to the rays traced through the first lens surface 
listed in Table 8-1, and give the values of the constant for h = 1.5, 1.0, 0.5, and cm. 

24. Apply Abbe's sine condition to the final rays traced through the lens in Table 
8-1 1, and give values of the constant for h = 1.5, 1.0, and 0.5 cm. 

Ans. 0.335270, 0.338143, and 0.339585. 

25. A thin lens of index 1.5 and radii r t = +40 cm and r t = —10 cm is used with 
parallel incident light. Calculate (a) the position factor, (ft) the shape factor, (c) 
the focal length, and (d) the longitudinal spherical aberration for rays at heights of 
h = 2.0 cm, 1.5 cm, 1.0 cm, and 0.5 cm. Plot a graph of h vs. longitudinal spherical 
aberration. 

26. A thin lens of index 1.50 and radii ft = +10 cm and r t = —40 cm is used 
with an object located 32 cm in front of the first surface. Calculate (a) the focal 
length, (ft) the position factor, (c) the shape factor, and (d) the longitudinal spherical 
aberration for rays at heights of 2.0, 1.5, 1.0, and 0.5 cm. 

Ans. (a) +16.0 cm. (ft) Zero, (c) +0.600. (d) 1.474 cm, 0.846 cm, 0.381 cm, 
0.196 cm. 

27. An achromatic lens with a focal length of +20 cm is to be made of crown and 
flint glasses of the types BSC and DF-4 (see Table 9-V). If the crown-glass lens 
is to be equiconvex and the combination is to be cemented, find (a) the v values, (ft) 
the two lens powers for sodium light, and (c) the radii of the four faces to correct 
for the C and F lines. 

28. An achromatic lens with a focal length of +12.5 cm is to be made of crown and 
flint glasses of the types LBC-1 and DF-2 (see Table 9-V). If the flint-glass lens 
is to have its outer face plane and the combination is to be cemented, find (a) the 
y-values, (ft) the two lens powers for sodium yellow light, and (c) the radii of the three 
remaining surfaces. The lens is to be corrected for the C and F lines. 

Ans. (a) 59.6472, 36.6172. (ft) +20.7198 D, -12.7198 D. (c) +5.6550 cm, 
-4.8507 cm, —4.8507 cm, infinity. 

29. An achromatic lens is to be made of BSC-2 and DF-4 glasses and is to have a 
focal length of +25 cm (see Table 9-V). If the flint-glass lens is to have its outer 
face plane and the combination is to be cemented, find the radii of curvature of the 
other three surfaces. The lens is to be corrected for the C and G' lines. 



168 GEOMETRICAL OPTICS 

30. An achromatic lens is to be made of SPC-1 and DF-2 glasses and is to have a 
focal length of +10 cm (see Table 9-V). If the crown-glass lens is to be equiconvex 
and the combination is to be cemented, what must be (a) the v values, (b) the powers 
of the two lenses for sodium light, and (c) the radii of curvature of the faces? The 
lens is to be corrected for the C and G' lines. 

Ans. (a) 37.5449, 22.7928. (b) +25.4505 D, -15.4505 D. (c) +4.1100 cm, 
-4.1100 cm, -4.1100 cm, +140.77 cm. 

31. Calculate the focal lengths of the lens in Prob. 28 for the C, D, F, and G' lines. 

32. Calculate the focal lengths of the lens in Prob. 30 for the C, D, F, and G' lines. 

Ans. (a) +10.0044 cm, +10.0000 cm, +9.9927 cm, +10.0044 cm. 



CHAPTER 10 

OPTICAL INSTRUMENTS 



The design of efficient optical instruments is the ultimate purpose of 
geometrical optics. The principles governing the formation of images 
by a single lens, and occasionally by simple combinations of lenses, have 
been set forth in the previous chapters. These principles find a wide 
variety of applications in the many practical combinations of lenses, 
frequently including also mirrors or prisms, which fall in the category of 
optical instruments. This subject is one of such large scope, and has 




Object 



-Image 



■Film 



Fig. KM. Principle of a camera. 

developed so many ramifications, that in a book devoted to the funda- 
mentals of optics it is only possible to describe the principles involved in 
a few standard types of instrument. In this chapter a description will be 
given of the more important features of camera lenses, magnifiers, micro- 
scopes, telescopes, and oculars. These will serve to illustrate some appli- 
cations of the basic ideas already discussed and will, it is hoped, be of 
interest to the student who has used, or expects to use, some of these 
instruments. 

10.1. Photographic Objectives. The fundamental principle of the 
camera is that of a positive lens forming a real image, as shown in Fig. 
10j4. Sharp images of distant or nearby objects are formed on a photo- 
graphic film or plate, which is later developed and printed to obtain 

169 



170 



GEOMETRICAL OPTICS 



the final picture. Where the scene to be taken involves stationary 
objects, the cheapest camera lens may, if it is stopped down almost 
to a pinhole and a time exposure used, yield photographs of excellent 
definition. If, however, the subjects are moving relative to the cam- 
era (and this includes the case where the camera is held in the hand), 
extremely short exposure times are often imperative and lenses of large 
aperture become a necessity. The most important feature of a good 
camera, therefore, is that it be equipped with a lens of high relative aper- 
ture capable of covering as large an angular field as possible. Because 

Exit 
pupil 

J Entrance 
pupil 




Fig. 1023. (a) Geometry for determining the speed of a lens, (b) An achromatic 
meniscus lens with a front stop. 



a lens of large aperture is subject to many aberrations, designers of photo- 
graphic objectives have resorted to the compromises as regards correction 
that best suit their particular needs. It is the intention here, therefore, 
to discuss briefly some of these purposes and compromises in connection 
with a few of the hundreds of well-known makes of photographic objective. 
10.2. Speed of Lenses. It was shown in Sec. 7.15 that the total 
amount of light reaching the image per unit area is given by the product 
of the brightness B of the source and the solid angle u' of the bundle of 
rays converging toward any point on the image. The latter may be 
computed as the area of the entrance pupil divided by the square of the 
focal length /. This will be clear from Fig. 10fi(a), which shows the 
lens and stop of Fig. 10.A illuminated by a parallel bundle. The solid 
angle to' is that subtended at the image point by the exit pupil, but as 
will be seen, this is equal to that which would be subtended by the 
entrance pupil if it were placed at the secondary principle plane H' . The 
ratio of the focal length of any lens to the linear diameter a of its entrance 
pupil is called its focal ratio, or / value, which is therefore defined as 



/ value = - 
a 



(10a) 



• 



OPTICAL INSTRUMENTS 171 

Thus a lens which has a focal length of 10 cm and a linear aperture of 
2 cm is said to have an / value of 5, or as it is usually stated, the lens 
is an //5 lens. 

The rapidity with which the photographic image is built up depends 
on the illuminance E of the image, which therefore determines the speed 
of the lens. The speed is inversely proportional to the square of the 
/ value, since by Eq. To, 

rr r» i D ""(a/2) 2 a 2 const. ,„... 

E = B„ cm B -p- - const. X j = y^j^p 006) 

assuming an object of a given brightness. 

In order to take pictures of faintly illuminated subjects, or of ones which 
are in rapid motion and require a very short exposure, a lens of small 
/ value is required. Thus an f/2 lens is "faster" than an //4.5 lens (or 
than an f/2 lens stopped down to//4.5) in the ratio (4.5/2) 2 = 5.06. A 
lens of such large relative aperture is difficult to design, as we shall see. 

10.3. Meniscus Lenses. Many of the cheapest cameras employ a sin- 
gle positive meniscus lens with a fixed stop such as was shown in Fig. 
10^4. Developed in about 1812 and called a landscape lens, this simple 
optical device exhibits considerable spherical aberration, thereby limiting 
its useful aperture to about //ll. Off the lens axis, the astigmatism 
limits the field to about 40°. The proper location of the stop results in 
a flat field, but with only a single lens there is always considerable 
chromatic aberration. 

By using a cemented doublet as shown in Fig. l0B(b), lateral chro- 
matism can be corrected. Instead of correcting for the C and F lines 
of the spectrum, however, the combination is usually corrected for the 
yellow D line, near the peak sensitivity of the eye, and the blue G' line, 
near the peak sensitivity of many photographic emulsions. Called " DG 
achromatism," this type of correction produces the best photographic 
definition at the sharpest visual focus. In some designs the lens and 
stop are turned around as in the arrangement of Fig. 9 £/(&). 

10.4. Symmetrical Lenses. Symmetrical lenses consist of two identi- 
cal sets of thick lenses with a stop midway between them; a number of 
these are illustrated in Fig. 10C(a). In general, each half of the lens is 
corrected for lateral chromatic aberration, and by putting them together, 
curvature of field and distortion are eliminated, as was explained in 
Sec. 9.11. In the rapid rectilinear lens, flattening of the field was made 
possible only by the introduction of considerable astigmatism, while 
spherical aberration limited the aperture to about //8. By introducing 
three different glasses, as in the Goerz "Dagor," each half of the lens 
could be corrected for lateral color, astigmatism, and spherical aberration. 
When combined they are corrected for coma, lateral color, curvature, and 
distortion. Zeiss calls this lens a "Triple Protar," while Goerz calls it 



- 



172 



GEOMETRICAL OPTICS 



the "DAGor," signifying Double Anastigmat Goerz. The "Speed Pan- 
chro" lens developed by Taylor, Taylor, and Hobson in 1920 is note- 
worthy because of its fine central definition combined with the high speed 
of f/2 and even //l. 5. The "Zeiss Topogon" lens is but one of a number 
of special "wide-angle" lenses, particularly useful in aerial photography. 
Additional characteristics of symmetrical lenses are (1) the large number 



Rapid rectilinear 
f/8 




Taylor, Taylor and Hobson 
Speed Poncro f/2 






Goerz "Dagor" f/4.5 Zeiss "Topogon' 




Original " Cooke Triplet" Zeiss "Tessor" 

Fia. IOC Symmetrical and unsymmetrical camera lenses. 



of lenses employed, and (2) the rather deep curves, which are expensive 
to produce. 

The greater the number of free glass surfaces in a lens, the greater is 
the amount of light lost by reflection. The / value alone, therefore, is 
not the sole factor in the relative speeds of objectives. The development 
in recent years of lens coatings that practically eliminate reflection at 
normal incidence has offered greater freedom in the use of more elements 
in the design of camera lenses (see Sec. 14.6). 

10.6. Triplet Anastigmats. A great step forward in photographic lens 
design was made in 1893 when H. D. Taylor of Cooke and Sons developed 
the "Cooke Triplet" (Fig. IOC). The fundamental principles involved 
in this system follow from the fact that (1) the power which a given lens 
contributes to a system of lenses is proportional to the height at which 
marginal rays pass through the lens, whereas (2) the contribution each 



OPTICAL INSTRUMENTS 173 

lens makes to field curvature in proportional to the power of the lens 
regardless of the distance of the rays from the axis. Hence astigmatism 
and curvature of field can be eliminated by making the power of the 
central flint element equal and opposite to the sum of the powers of the 
crown elements. By spacing the negative lens between the two positive 
lenses, the marginal rays can be made to pass through the negative lens 
so close to the axis that the system has an appreciable positive power. A 
proper selection of dispersions and radii enables additional corrections 
to be made for color and spherical aberration. The "Tessar," one of the 
best known modern photographic objectives, was developed by Zeiss in 
1902. Made in many forms to meet various requirements, the system has 




Fig. 10D. Principle of the telephoto lens. 

a general structure similar to that of a Cooke Triplet in which the rear 
crown lens is replaced by a doublet. The Leitz "Hector," working at 
f/2, is also of the Cooke Triplet type, but each element is replaced by a 
compound lens. This very fast lens is excellent in a motion-picture 
camera. 

10.6. Telephoto Lenses. Since the image size for a distant object is 
directly proportional to the focal length of the lens, a telephoto lens which 
is designed to give a large image is a special type of objective with a longer 
effective focal length than that normally used with the same camera. 
Because this would require a greater extension of the bellows than most 
cameras will permit, the principle of a single highly corrected thick lens 
is modified as follows: As is shown in Fig. 10D by the refraction of an 
incident parallel ray, with two such lenses considerably separated the 
principal point H' can be placed well in front of the first lens, thereby 
giving a long focal length H'F' with a short lens-to-focal-plane distance 
(f b in Fig. 10D). The latter distance, or the back focal length as it is 
usually called, is measured from the real lens to the focal plane, as shown. 

Although the focal lengths of older types of telephoto lenses could be 
varied by changing the distance between the front and rear elements, 
these lenses are almost always made with a fixed focal length. Flexibility 
is then obtained by having a set of lenses. This has become necessary 
through the desire for lenses of greater speed and better correction of the 



174 



GEOMETRICAL OPTICS 



aberrations. A "Cooke Telephoto" as produced by Taylor, Taylor, and 
Hobson is shown in Fig. 10E. 

10.7. Magnifiers. The magnifier is a positive lens whose function it 
is to increase the size of the retinal image over and above that which is 
formed with the unaided eye. The apparent size of any object as seen 
with the unaided eye depends on the angle subtended by the object (see 

Fig. 10F). As the object is brought closer 
to the eye, from A to I? to C in the dia- 
gram, accommodation permits the eye to 
change its power and to form a larger and 
larger retinal image. There is a limit to 
how close an object may come to the eye 
\ I \ mm if the latter is still to have sufficient ac- 

commodation to produce a sharp image. 
Although the nearest point varies widely 
with various individuals, 25 cm is taken 
to be the standard near point, or as it is sometimes called the distance 
of most distinct vision. At this distance, indicated in Fig. 10G(a), the 
angle subtended by object or image will be called 0. 

If a positive lens is now placed before the eye in the same position, as 
in diagram (b), the object y may be brought much closer to the eye and 
an image subtending a larger angle 6' will be formed on the retina. What 




Axis 



Fig. 102?. A well-corrected tele- 
photo lens. 




Fig. 10F. The angle subtended by the object determines the size of the retinal image. 

the positive lens has done is to form a virtual image y' of the object y 
and the eye is able to focus upon this virtual image. Any lens used in 
this manner is called a magnifier or simple microscope. If the object y is 
located at F, the focal point of the magnifier, the virtual image y' will 
be located at infinity and the eye will be accommodated for distant vision 
as is illustrated in Fig. l(X?(c). If the object is properly located a short 
distance inside of F as in diagram (6), the virtual image may be formed 
at the distance of most distinct vision and a slightly greater magnification 
obtained, as will now be shown. 

The angular magnification M is defined as the ratio of the angle 6' 
subtended by the image to the angle subtended by the object. 



M = j 



(10c) 



OPTICAL INSTRUMENTS 



17* 



(a) 



(6) 





(0 



Fig. 10G. Illustrating the angle subtended by (a) an object at the near point to the 
naked eye, (6) the virtual image of an object inside the focal point, (c) the virtual image 
of an object at the focal point. 

From diagram (b), the object distance s is obtained by the regular thin- 
lens formula as 



1 



1 + 

s T -25 / 



or 



1 m 25+/ 
s 25/ 



From the right triangles, the angles 6 and 0' are given by 



tan = ^p and 
25 



-'" l s = » 2 W 



For small angles the tangents can be replaced by the angles themselves 
to give approximate relations 



= To and 



= y 



25+/ 
25/ 



giving for the magnification, from Eq. (10c), 



M = - = 7 + l 



(lOd) 






176 



GEOMETRICAL OPTICS 



In diagram (c) the object distance s is equal to the focal length, and the 
small angles and 6' are given by 



a - y 
"-25 



and 



giving for the magnification 



M = ^ = =Z 



25 



(10e) 



The angular magnification is therefore larger if the image is formed at the 
distance of most distinct vision. For example, let the focal length of a 
magnifier be 1 in. or 2.5 cm. For these two extreme cases, Eqs. lOd 
and lOe give 



^S +1 



= nx 



and 



M = 



25 
2.5 



- 10X 



Because magnifiers usually have short focal lengths and therefore give 
approximately the same magnifying power for object distances between 
25 cm and infinity, the simpler expression 25// is commonly used in 
labeling the power of magnifiers. Hence a magnifier with a focal length 
of 2.5 cm will be marked 10 X and another with a focal length of 5 cm 
will be marked 5X, etc. 

10.8. Types of Magnifiers. Several common forms of magnifiers are 
shown in Fig. 10H. The first, an ordinary double-convex lens, is the 





Double 
convex 






Doublet Coddington Hastings 

triplet 

Fig. 10//. Common types of magnifiers. 



Achromat 



simplest magnifier and is commonly used as a reading glass, pocket 
magnifier, or watchmaker's loupe. The second is composed of two 
identical plano-convex lenses each mounted at the focal point of the 
other. As shown by Eq. 9z this spacing corrects for lateral chromatic 
aberration but requires the object to be located at one of the lens faces. 
To overcome this difficulty, color correction is sacrificed to some extent 
by placing the lenses slightly closer together, but even then the working 
distance or back focal length (see Eq. 5m) is extremely short. 



OPTICAL INSTRUMENTS 



177 



Eyepiece 



The third magnifier, cut from a sphere of solid glass, is commonly 
credited to Coddington but was originally made by Sir David Brewster. 
It too has a relatively short working distance, as can be seen by the 
marginal rays, but the image quality is remarkably good due in part to 
the central groove acting as a stop. Some of the best magnifiers of 
today are cemented triplets, such as are 
shown in the last two diagrams. These 
lenses are symmetrical to permit their 
use either side up. They have a rela- 
tively large working distance and are 
made with powers up to 20 X . 

10.9. Microscopes. The microscope, 
which in general greatly exceeds the 
power of a magnifier, was invented by 
Galileo in 1610. In its simplest form, 
the modern optical microscope consists 
of two lenses, one of very short focus 
called the objective, and the other of 
somewhat longer focus called the ocular 
or eyepiece. While both these lenses 
actually contain several elements to re- 
duce aberrations, their principal func- 
tion is illustrated by single lenses in 
Fig. 107. The object (1) is located just 
outside the focal point of the objective 
so there is formed a real magnified image 
at (2). This image becomes the object 
for the second lens, the eyepiece. Func- 
tioning as a magnifier, the eyepiece forms a large virtual image at (3). 
This image becomes the object for the eye itself, which forms the final 
real image on the retina at (4). 

Since the function of the objective is to form the magnified image that 
is observed through the eyepiece, the overall magnification of the instru- 
ment becomes the product of the linear magnification mi of the objective 
and the angular magnification M 2 of the eyepiece. By Eqs. 4fc and lOe, 
these are given separately by 




Fig. 10/. Principle of the microscope, 
shown with the eyepiece adjusted to 
give the image at the distance of 
most distinct vision. 



mi = -7 and 
The over-all magnification is, therefore, 



Mi = -T- 



M = r • -s- 



/l h 



(10/) 



178 



GEOMETRICAL OPTICS 



It is customary among manufacturers to label objectives and eyepieces 
according to their separate magnifications Wi and M %. 

10.10. Microscope Objectives. A high-quality microscope is usually 
equipped with a turret nose carrying three objectives, each of a different 
magnifying power. By turning the turret, any one of the three objec- 
tives may be rotated into proper alignment with the eyepiece. Diagrams 
of three typical objectives are shown in Fig. 10/. The first, composed 
of two cemented achromats, is corrected for spherical aberration and 
coma and has a focal length of 1.6 cm, a magnification of 10 X, and a 




<&&* 





la) (6) (c) 

Fig. 10J. Microscope objectives, (a) Low-power, (b) medium-power, and (c) high- 
power oil immersion. 



working distance of 0.7 cm. The second is also an achromatic objective 
with a focal length of 0.4 cm, a magnification of 40 X, and a working 
distance of 0.6 cm. The third is an oil-immersion type of objective with 
a focal length of 0.16 cm, a magnification of 100 X, and a working distance 
of only 0.035 cm. Great care must be exercised in using this last type 
of lens to prevent scratching of the hemispherical bottom lens. Although 
oil immersion makes the two lowest lenses aplanatic (sec Fig. 90), lateral 
chromatic aberration is present. The latter is corrected by the use of a 
compensating ocular, as will be explained in Sec. 10.16. 

10.11. Astronomical Telescopes. Historically the first telescope was 
probably constructed in Holland in 1608 by an obscure spectacle-lens 
grinder, Hans Lippershey. A few months later Galileo, upon hearing 
that objects at a distance could be made to appear close at hand by 
means of two lenses, designed and made with his own hands the first 
authentic telescope. The elements of this telescope are still in existence 
and may be seen on exhibit in Florence. The principle of the astronomi- 
cal telescopes of today is the same as that of these early devices. A 
diagram of an elementary telescope is shown in Fig. 10K. Rays from 
one point of the distant object are shown entering a long-focus objective 



OPTICAL INSTRUMENTS 



179 



lens as a parallel beam. These rays are brought to a focus and form a 
point image at Q' . Assuming the distant object to be an upright arrow, 
this image is real and inverted as shown. The eyepiece has the same 
function in the telescope that it has in a microscope, namely, that of a 
magnifier. If the eyepiece is moved to a position where this real image 




Fig. \0K. Principle of the astronomical telescope, shown with the eyepiece adjusted 
to give the image at the distance of most distinct vision. 



Objective 



Eyepiece 



Final 
image 




Fig. 10L. Principle of the astronomical telescope, shown with the eyepiece adjusted to 
give the image at infinity. 



lies just inside its primary focal "lane F t , a magnified virtual image at 
Q" may be seen by the eye at the near point, 25 cm. Normally, however, 
the real image is made to coincide with the focal points of both lenses, 
with the result that the image rays leave the eyepiece as a parallel bundle 
and the virtual image is at infinity. The final image is always the one 
formed on the retina by rays which appear to have come from Q". 
Figure 10L is a diagram of the telescope adjusted in this manner. 

In all astronomical telescopes the objective lens is the aperture stop. 
It is therefore the entrance pupil, and its image as formed by all the 
lenses to its right (here, only the eyepiece) is the exit pupil. These 



180 



GEOMETRICAL OPTICS 



elements are shown in Fig. 10M, which traces the path of one ray incident 
parallel to the axis and of a chief ray from a distant off-axis object point. 
The distance from the eye lens, i.e., the last lens of the ocular, to the exit 
pupil is called the eye relief and should normally be about 8 mm. 



Entrance 
pupil 




Fig. 10A/. Entrance and exit pupils of an astronomical telescope. 

The magnifying power of a telescope is defined as the ratio between 
the angle subtended at the eye by the final image Q" and the angle sub- 
tended at the eye by the object itself. The object, not shown in Fig. 
10M, subtends an angle at the objective and would subtend approxi- 
mately the same angle to the unaided eye. The angle subtended at the 
eye by the final image is the d' . By definition, 



M = - 



(I0g) 



The angle 6 is the object-field angle, and 6' is the image-field angle. In other 
words, 6 is the total angular field taken in by the telescope while 6' is the 
angle that, the field appears to cover (Sec. 7.11). From the right triangles 
ABC and EBC, in Fig. lOilf , 



tan 6 = - 
s 



and 



tan 0' = 7 

8 



Applying the general lens formula 1/s -+- 1/s' = 1//, 



1 



So 



s' JeUo+Se) 
which, substituted in Eq. lO/i, gives 
h 



(lO/i) 



(ioo 



tan 6 = 



Uo+fm) 



and tan 6' = — 



hfo 



SbVo+Sb) 



For small angles, tan c^ d and tan 6' ~ 0' . Substituting them in 
Eq. 10<7, we obtain 



]E 



(10/) 



OPTICAL INSTRUMENTS 181 

Hence the magnifying power of a telescope is just the ratio of the focal 
lengths of objective and eyepiece respectively, the minus sign signifying 
an inverted image. 

If D and d represent the diameters of the objective and exit pupil 
respectively, the marginal ray passing through F' and F E in Fig. 10.1/ 
forms two similar right triangles, from which the following proportion is 
obtained 

Sb d 

giving, as an alternative equation for the angular magnification, 

M = § (10*) 

A useful method of determining the magnification of a telescope is, 
therefore, to measure the ratio of the diameters of the objective lens and 
of the exit pupil. The latter is readily found by focusing the telescope 
for infinity and then turning it toward the sky. A thin sheet of white 
paper held behind the eyepiece and moved back and forth will locate a 
sharply defined disk of light. This, the exit pupil, is commonly called 
the Ramsden circle. Its size, relative to that of the pupil of the eye, is 
of great importance in determining the brightness of the image and the 
resolving power of the instrument (see Sees. 7.15 and 15.9). 

Another method of measuring the magnification of a telescope is to 
sight through the telescope with one eye, observing at the same time 
the distant object directly with the other eye. With a little practice 
the image seen in the telescope can be made to overlap the smaller direct 
image, thereby affording a straightforward comparison of the relative 
heights of image and object. The object field of the astronomical tele- 
scope is determined by the angle subtended at the center of the objective 
by the eyepiece aperture. In other words, the eyepiece is the field stop 
of the system. In Fig. 10M the angle 9 is the half-field angle (Sec. 7.8). 

10.12. Oculars or Eyepieces. Although a simple magnifier of one of 
the types shown in Fig. lOif may be used as an eyepiece for a microscope 
or telescope, it is customary to design special lens combinations for each 
particular instrument. Such eyepieces are commonly called oculars. 
One of the most important considerations in the design of oculars is 
correction for lateral chromatic aberration. It is for this reason that the 
basic structure of most of them involves two lenses of the same glass and 
separated by a distance equal to half the sum of their focal lengths 
(see Eq. 9z). 

The two most popular oculars based on this principle are known as the 
Huygens eyepiece and the Ramsden eyepiece (Fig. ION). In both these 



182 



GEOMETRICAL OPTICS 



systems the lens nearest the eye is called the eye lens, while the lens 
nearest the objective is called the field lens. 

10.13. Huygens Eyepiece. In eyepieces of this design the two lenses 
are usually made of spectacle crown glass with a focal-length ratio ////« 
varying from 1.5 to 3.0. As shown in Fig. 10JV(a), rays from an objec- 
tive to the left (and not shown) are converging to a real image point Q. 
The field lens refracts these rays to a real image at Q', from which they 
diverge again to be refracted by the eye lens into a parallel beam. In 
most telescopes the objective of the instrument is the entrance pupil of 
the entire system. The exit pupil or eyepoint is, therefore, the image of 




Ramsden eyepiece 

(b) 
Fig. ION. Common eyepieces used in optical instruments. 

the objective formed by the eyepiece and is located at the position 
marked "Exit pupil " in the figure. Here the chief ray crosses the axis of 
the ocular. A field stop FS is often located at Q', the primary focal 
point of the eye lens, and if cross hairs or a reticle are to be employed, 
they are mounted at this point. Although the eyepiece as a whole is 
corrected for lateral chromatic aberration, the individual lenses are not, 
so that image of the cross hairs or reticle formed by the eye lens alone will 
show considerable distortion and color. Huygens eyepieces with reticles 
are used in some microscopes, but in this case the reticle is small and is 
confined to the center of the field. The Huygens eyepiece shows some 
spherical aberration, astigmatism, and a rather large amount of longi- 
tudinal color and pincushion distortion. In general, the eye relief — i.e., 
the distance between the eye lens of the ocular and the exit pupil — is 
too small for comfort. 

10.14. Ramsden Eyepiece. In eyepieces of this type as well, the two 
lenses are usually made of the same kind of glass, but here they have equal 
focal lengths. To correct for lateral color, their separation should be 
equal to the focal length. Since the first focal plane of the system coin- 
cides with the field lens, a reticle or cross hairs must be located there. 
Under some conditions this is considered desirable, but the fact that any 
dust particles on the lens surface would also be seen in sharp focus is an 
undesirable feature. To overcome this difficulty, the lenses are usually 



OPTICAL INSTRUMENTS 



183 



moved a little closer together, thus moving the focal plane forward at 
some sacrifice of lateral achromatism. 

The paths of the rays through a Ramsden eyepiece as shown in Fig. 
10iV(6). The image formed by an objective (not shown) is located at the 
first focal point F, and it is here that a field stop FS and a reticle or 
cross hairs are often located. After refraction by both lenses, parallel 
rays emerge and reach the eye at or near the exit pupil. With regard to 
aberrations, the Ramsden eyepiece has more lateral color than the 
Huygens eyepiece but the longitudinal color is only about half as great. 
It has about one-fifth the spherical aberration, about half the distortion, 
and no coma. One important advantage over the Huygens ocular is its 
50 per cent greater eye relief. 

10.15. Kellner or Achromatized Ramsden Eyepiece. Because of the 
many desirable features of the Ramsden eyepiece, various attempts have 
been made to improve its chromatic defects. This aberration can be 





Kellner achromatized 
Ramsden eyepiece 



Orthoscopic 
eyepiece 



Symmetrical 
eyepiece 



Fig. 10O. Three types of achromatized eyepiece. 

almost eliminated by making the eye lens a cemented doublet (Fig. 100). 
Such eyepieces are commonly used in prism binoculars, because the 
slight amount of lateral color is removed and spherical aberration is 
reduced through the aberration characteristics of the Porro prisms 
(Sec. 2.2). 

10.16. Special Eyepieces. The orthoscopic eyepiece shown in the mid- 
dle diagram of Fig. 10O is characterized by its wide field and high mag- 
nification. It is usually employed in high-power telescopes and range 
finders. Its name is derived from the freedom from distortion character- 
izing the system. The symmetrical eyepiece shown at the right in Fig. 
100 has a larger aperture than a Kellner of the same focal length. This 
results in a wider field as well as a long eye relief, hence its frequent use 
in various types of telescopic gun sights. The danger of having a short 
eye relief with a recoiling gun should be obvious. 

Since lateral chromatic aberration, as well as the other aberrations of 
an eyepiece, is affected by altering the separation of the two elements, 
some oculars are provided with means for making this distance adjust- 
able. Some microscopes come equipped with a set of such compensating 
eyepieces, thereby permitting the undercorrection of lateral color in any 
objective to be neutralized by an overcorrection of the eyepiece. 



184 



GEOMETRICAL OPTICS 



10.17. Prism Binoculars. Prism binoculars are in reality a pair of 
identical telescopes mounted side by side, one for each of the two eyes. 
Such an instrument is shown in Fig. 10P with part of the case cut away 
to show the optical parts. The objectives are cemented achromatic 
pairs, while the oculars are Kellner or achromatized Ramsden eyepieces. 
The dotted lines show the path of an axial ray through one pair of Porro 
prisms. The first prism reinverts the image and the second turns it left 
for right, thereby finally giving an image in the proper position. The 




Fig. 107'. Diagram of prism binoculars, showing the lenses and totally reflecting Porro 
prisms. 

doubling back of the light rays has the further advantage of enabling 
longer focus objectives to be used in short tubes, with consequent higher 
magnification. 

There are four general features that go to make up good binoculars: 
(1) magnification, (2) field of view, (3) light-gathering power, and (4) size 
and weight. For hand-held use, binoculars with five-, six-, seven-, or 
eightfold magnification are most generally used. Glasses with powers 
above 8 are desirable, but require a rigid mount to hold them steady. 
For powers less than 4, lens aberrations usually offset the magnification, 
and the average person can usually see better with the unaided eyes. 
The field of view is determined by the eyepiece aperture and should be as 
large as is practicable. For seven-power binoculars a 6° object field is 
considered large, since in the eyepiece the same field is spread over 
an angle of 7 X 6°, or 42°. 

The diameter of the objective lenses determines the light-gathering 
power. Large diameters are important only at night when there is little 



OPTICAL INSTRUMENTS 



185 




Corrector 
lens 

Fig. 10Q. 
system. 



Kellner-Schmidt optical 



light available. Binoculars with the specification 6 X 30 have a magni- 
fication of 6 and objective lenses with an effective diameter of 30 mm. 
The specification 7 X 50 means a magnification of 7 and objectives 
50 mm in diameter. Although glasses with the latter specifications are 
excellent for day or night use, they are considerably larger and more 
cumbersome than the daytime glasses specified as 6 X 30 or 8 X 30. 
For general civilian use, the latter two are much the most useful. 

10.18. The Kellner-Schmidt Optical System. The Kellner-Schmidt 
optical system combines a concave spherical mirror with an aspheric 
lens as shown in Fig. 10Q. Kellner 
devised and patented* this optical 
system in 1910 as a high-quality source 
of parallel light. Years later Schmidt 
introduced the system as a high-speed 
camera, and it has since become known 
as a Schmidt camera. While Schmidt 
was the first to emphasize the impor- 
tance of placing the corrector plate at 
the mirror's center of curvature, Kellner 
shows it in this position in his patent 
drawing. 

The purpose of the lens is to refract incoming parallel rays in such 
directions that after reflection from the spherical mirror they all come 
to a focus at the same axial point F. This "corrector plate," there- 
fore, eliminates the spherical aberration of the mirror. With the lens 
located at the center of curvature of the mirror, parallel rays entering 
the system at large angles with the axis are brought to a relatively good 
focus at other points like F' . The focal surface of such a system is 
spherical, with its center of curvature at C. 

Such an optical system has several remarkable and useful properties. 
First as a camera, with a small film at the center or with a larger film 
curved to fit the focal surface, it has the very high speed of //O.5. Because 
of this phenomenal speed, Schmidt systems are used by astronomers to 
obtain photographs of faint stars or comets. They are used for similar 
reasons in television receivers to project small images from an oscilloscope 
tube into a relatively large screen. In this case the convex oscilloscope 
screen is curved to the focal surface so that the light from the image 
screen is reflected by the mirror and passes through the corrector lens 
to the observing screen. 

If a convex silvered mirror is located at FF', rays from any distant 
source will on entering the system form a point image on the focal surface 
and after reflection will again emerge as a parallel bundle in the exact 

* American Patent No. 969,785, 1910. 



186 



GEOMETRICAL OPTICS 



direction of the source. When used in this manner the device is called an 
autocollimator . If the focal surface is coated with fluorescent paint, ultra- 
violet light from a distant invisible source will form a bright spot at some 
point on FF' , and the visible light emitted from this spot will emerge only 
in the direction of the source. If a hole is made in the center of the large 
mirror, an eyepiece may be inserted in the rear to view the fluorescent 
screen and any ultraviolet source may be seen as a visible source. As 
such, the device becomes a fast, wide-angled, ultraviolet telescope. 

10.19. Concentric Optical Systems.* The recent development and use 
of concentric optical systems warrants at least some mention of their 




Fig. \0R. Concentric optical system. 

remarkable optical properties. Such systems have the general form of a 
concave mirror and a concentric lens of the type shown in Fig. 5/. As 
the title implies, and as is shown in Fig. 10R, all surfaces have a common 
center of curvature C. 

The purpose of the concentric lens is to reduce spherical aberration 
to a minimum. Off-axis rays traversing the lens are bent away from 
the axis and, by the proper choice of radii, refractive index, and lens 
thickness, can be made to cross the axis at the paraxial focal point F. 
Since any ray through C may be considered as an axis, the focal surface 
is also a sphere with C as a center of curvature. In some applications the 
back surface of the lens is made to be the focal surface. 

Since the principal planes of the concentric lens both coincide with a 
plane through C perpendicular to the axial ray of any bundle, it is as if 

* A. Bouwers, "Achievements in Optics," Elsevier Press, Inc., Houston, Tex., 1950. 



OPTICAL INSTRUMENTS 187 

the corrector were a thin lens located at C and oriented at the proper 
angle for all incident parallel beams. 

Since there are no oblique and no sagittal rays, the system is free of 
coma and astigmatism. The complete performance of the system is 
known as soon as the imagery of an axial object point is known. Here 
lies the essential advantage over the Kellner-Schmidt system. Chromatic 
aberrations resulting from the lens are small as long as the focal length 
is long compared with that of the mirror, and this is nearly always the 
case. 

Other important features of the concentric system may be seen from 
the diagram. There is an unusually small decrease in image brightness 
with increasing angle of incidence. The corrector lens may be placed 
in front of C, in position 2. In this position the same optical performance 
is realized. Finally, a concentric convex mirror may be placed about 
halfway between the lens and the mirror. The reflected light is then 
brought to a focus through a hole in the center of the large mirror. This 
latter arrangement, among other things, makes an excellent objective 
system for a reflecting microscope. 

PROBLEMS 

1. A magnifier is made of two thin plano-convex lenses, each with a 2-cm focal 
length, and spaced 1.5 cm apart. Applying Gaussian formulas, find (a) its focal 
length, (b) its magnifying power, and (c) its back focal length. 

2. A Coddington magnifier made from a sphere of 1.5 cm diameter is made of crown 
glass of index 1.50. Find by calculation (a) its focal length, (b) its magnifying power, 
and (c) its back focal length. Ans. (a) + 1.12 cm. (6) 22.2 X. (c) 0.375 cm. 

3. A Ramsden eyepiece is made of two thin plano-convex lenses, each of 2.5 cm 
focal length, and spaced 1.8 cm apart. Applying thin-lens formulas, find (a) its 
focal length, (6) its magnifying power, and (c) its back focal length. 

4. A Ramsden eyepiece is made of two thin lenses, each with a 22.5-mm focal 
length, and spaced 16 mm apart. Applying thin-lens formulas, find (a) its focal 
length, (b) its magnifying power, and (c) its back focal length. 

Ans. (a) -f- 1.745 cm. (6) 14.3 X. (c) 5.05 mm. 

5. A Huygens eyepiece is made of two thin lenses, with focal lengths of 2 cm and 
1 cm, respectively. If the lenses are spaced to correct for chromatic aberration, find 
(a) the focal length of the combination, (6) the magnifying power, and (c) the back 
focal length. 

6. A microscope has an objective with a focal length of 3 mm and an ocular marked 
20 X. What is the total magnification if the objective forms its image 16 cm beyond 
its secondary focal plane? Ans. 1067 X. 

7. A microscope has an objective with a focal length of 3.5 mm and an ocular 
with a focal length of 10 mm. What is the total magnification if the objective forms 
its image 16 cm beyond its secondary focal plane? 

8. The eyepiece and objective of a microscope are 20.6 cm apart, and each has a 
focal length of 6 mm. Treating these lenses as though they were thin lenses, find (a) 
the distance from the objective to the object viewed, (6) the linear magnification 



188 GEOMETRICAL OPTICS 

produced by the objective, and (c) the over-all magnification if the final image is 
formed at infinity. Ans. (a) 6.19 mm. (6) -32.3 X. (c) 1347X. 

9. The ocular and objective of a microscope are 21.2 cm apart, and each has a 
focal length of 1.2 cm. Treating these as thin lenses, find (a) the distance from the 
objective to the object viewed, (b) the linear magnification produced by the objective, 
and (c) the over-all magnification if the final image is formed at infinity. 

10. An objective of an astronomical telescope has a diameter of 10 cm and a focal 
length of 120 cm. When it is used with an eyepiece having a focal length of 2 cm 
and a field lens with a diameter of 10 mm, find (a) the angular magnification, (6) the 
diameter of the exit pupil, (c) the object-field angle, (d) the image-field angle, and (e) 
the eye relief. Ans. (a) 60 X. (b) 1.67 mm. (c) 0.47°. (d) 28.2°. (e) 2.03 cm. 

11. An astronomical telescope has an objective with a diameter of 6 cm and a focal 
length of 60 cm. When used with a 20 X ocular having a field lens 1.2 cm in diameter, 
find (a) the angular magnification, (b) the diameter of the exit pupil, (c) the object-field 
angle, and (d) the image-field angle. 

12. The objectives of a pair of binoculars have apertures of 60 mm and focal lengths 
of 250 mm. The oculars have apertures of 10 mm and focal lengths of 22 mm. Find 
(a) the angular magnification, (6) the diameter of the exit pupils, (c) the object-field 
angle, (d) the image-field angle, (e) the eye relief, and (/) the field at 1000 m. 

Ans. (a) 11.4 X. (6) 5.28 mm. (c) 2.1°. (d) 24.5°. (e) 2.4 cm. (/) 36.8 m. 



PART II 

PHYSICAL OPTICS 



CHAPTER 11 

LIGHT WAVES 



The preceding chapters were concerned with the subject of geometrical 
optics, the basis of which is furnished by the laws of reflection and refrac- 
tion. We now turn to physical optics, which comprises those phenomena 
bearing on the nature of light. As thus defined, this field includes proc- 
esses which involve the interactions of light with matter, as for example 
the emission and absorption of light. Many of these processes require 
the quantum theory for their complete explanation, but the systematic 
treatment of this theory lies beyond the scope of this book. A large and 
homogeneous class of optical phenomena can be explained by assuming 
that light consists of waves, and it has therefore seemed desirable to 
restrict the meaning of the term "physical optics" to include only the 
classical wave theory of light. The way in which this theory forms part 
of the more complete one called quantum mechanics will then be briefly 
described in the final chapter (Chap. 30). 

As we have seen, large-scale optical effects can be explained by the use 
of light rays. Finer details require the wave picture which we are now 
to consider. Most of these details are not commonly observed in every- 
day life but appear when, for example, we make a close examination of 
the effects of passing light through narrow openings or of reflecting it 
from ruled surfaces. Finally, processes which occur on a still smaller 
scale, involving individual atoms or molecules, require quantum theory 
for their rigorous treatment. Any case of the interaction of two or more 
beams of light with each other may, however, be treated quantitatively 
by wave theory. As an introduction to this theory, the present chapter 
deals with wave motion in general and indicates at appropriate points 
how the various characteristics of light depend on those of the waves of 
which we assume it to consist. 

11.1. Wave Motion. Waves of the type with which we are most 
familiar, i.e., waves on the surface of water, are of considerable com- 
plexity. However, they may serve to illustrate an important feature 
present in any wave motion. If the waves are traveling in the x direc- 
tion and the y direction is vertical, an instantaneous picture of the con- 
tour of the waves in the x,y plane is given in Fig. 11.4 by the continuous 

191 



192 



PHYSICAL OPTICS 



curve. Let this curve be represented by an equation y = f(x). If the 
wave contour is to move toward -\-x with a constant velocity v, we must 
introduce the time t in such a way that, as t increases, a given ordinate 
such as ?/i will, after a time At has elapsed, be found at y[, farther to the 
right by an amount Ax = v At. This is accomplished by writing the 
equation y = f(x — vt), since we have, at the two times t and t + At, 

Vi = f(x - vt) 

y[ = f[(x + Ax) - v(t + AO] 

If now we substitute Ax = v At, we find that y[ = y\, and the above 
requirement is realized. The wave is in the position of the broken curve 



t t+At JL 



-y 




Fig. 11.4. Illustrating the propagation of water waves. 



at the instant t -f- At. The general equation for any transverse wave 
motion in a plane is 

y =f(x± vt) (11a) 

The plus sign is to be used if the wave is to travel to the left, i.e., in the 
— x direction. 

This equation is the solution of the so-called wave equation, a partial 
differential equation which for waves traveling along the x axis may be 
written 



djy 
dt 2 



= v 



d*y 
dx 2 



(116) 



To prove that Eq. 11a satisfies it, we evaluate the derivatives, using, for 
example, the negative sign. Partial differentiations with respect to t give 



and 



|f = -vf'(x - vt) 
|^ - v 2 f"(x - vt) 



while the differentiations with respect to x give 



dx 



M - /"(* - »o 



LIGHT WAVES 



193 




The proportionality factor v- is therefore obtained. Now the second 
derivative with respect to t represents the acceleration of a particle 
at a given instant, while that with respect to x determines the curvature 
of the wave contour at the same point and instant. Thus, if one can 
evaluate these derivatives for a given kind of waves, he has a means of 
finding their velocity. This method will be used in Sec. 11.4 for trans- 
verse elastic waves and again in Sec. 20.4 for electromagnetic waves. 

The above equations represent the progression of the wave contour 
with time and specify that, whatever the initial form, the form at time / 
is the same but is displaced by a 
distance vt. This does not imply _ 

that the particles of the medium 
are carried along with the wave. 
On the contrary, the only thing that 
moves along continuously is the 
contour, while each particle merely 
oscillates about its position of equi- 
librium. Nor do the equations 
set any restriction on the type of 
oscillations involved. In water 
waves, for example, they are cir- 
cular or elliptical ones in the x,y 
plane of Fig. 11 A. This figure, of 
course, represents merely a cross section perpendicular to the crests of 
the waves. The complete waves are spread over the x,z plane, and their 
crests are straight since the displacement y in Eq. 11a is independent of z. 

Turning now from surface waves to waves in three dimensions such as 
sound waves or earthquake waves, the same equations may be applied. 
For this to be so, it is necessary that the locus of equal displacements 
occur in a plane, and we speak of plane waves. Such waves could be 
produced in a block of elastic material, for example, as is illustrated in 
Fig. IIB. A board attached to one surface of the block will, if given a 
periodic motion in its own plane, generate plane waves. Equations 11a 
and 116 will represent such waves if the perpendicular to the wave fronts 
is parallel to x. To generalize the equations so that they can represent 
plane waves going in any direction, one merely substitutes for x the 
expression Ix + my + nz, where I, m, and n are the cosines of the angles 
between the given direction and the x, y, and z axes, respectively. 

A small source of light generates waves which are not plane but 
spherical, with the source at the center of the curved wave fronts. 
Since the curvature decreases with distance, one may realize what are 
essentially plane waves of light by placing the source sufficiently far away. 
The required distance depends on the aperture, i.e., on the area of the 



Fig. 11B. Generation of transverse waves 
in an elastic solid. 



194 



PHYSICAL OPTICS 



wave front used, the distance obviously increasing for larger apertures. 
A more common way of obtaining plane light waves is to place a point 
source at the primary focal point of a lens or mirror, as was shown in 
Chaps. 4 and 6. It is true that in practice the source is never a mathe- 
matical point and that the beam obtained actually consists of many 
plane waves slightly inclined to each other, each originating from a 
different point on the source. To reduce this effect to a minimum, the 
usual laboratory practice is to employ as the point source an illuminated 
pinhole of diameter not exceeding a few wavelengths of light. 

11.2. Sine Waves. The simplest type of wave is that for which the 
function / in Eq. 11a is a sine or cosine. The motion of the individual 
particles is then simple harmonic.* This is the type of motion that one 
expects for elastic substances, where the forces due to distortion obey 
Hooke's law. Let us consider transverse waves, in which the motions of 
the particles are perpendicular to the direction of travel of the waves. 
The instantaneous displacements y may then be represented by writing 



y = a sin 



2ttx 



The significance of the constants a and X may be seen from the curve 



1 





a 


— -i 


PI 








I 




P ">\ 


■4 


/* 


3 V\*^ 









. 









Fig. 11C. Contour of a sine wave at time t = 0. 

of Fig. 11C, which is a plot of the above equation. The maximum dis- 
placement a is called the amplitude of the wave, and the distance X after 
which the curve repeats, its wavelength. 

To represent the wave in time as well as in space, i.e., to make the 
wave move, one introduces the time as in Eq. 11a, obtaining 



y = a sin — (x — vt) 



(He) 



Then the contour will be displaced toward +x with the velocity v. 
Any one particle, such as P in the figure, will carry out a simple harmonic 

* For a discussion of simple harmonic motion, and of its mathematical representa- 
tion, the reader may refer to any textbook on elementary mechanics, such as F. W. 
Sears, "Principles of Physics," vol. I, Mechanics, Heat and Sound, 2d ed., Addison- 
Wesley Publishing Company, Cambridge, Mass., 1946. 



LIGHT WAVES 195 

vibration, occupying the successive positions P, P', P", etc., as the wave 
moves. The time for a whole vibration of any particle is the period T, 
and its reciprocal, the number of vibrations per second, is the frequency v. 
We have 

t>=^= vk (lid) 

A useful and concise way of expressing the equation for simple har- 
monic waves is in terms of the angular frequency u> = 2irv and the propaga- 
tion number* k = 2v/\. Equation lie then becomes 

y = a sin (kx — at) = a sin (ut — kx + ir) 
= a cos ( (at — kx + n ) 

Now the addition of a constant to the quantity in parentheses is of 
little physical significance, since such a constant may be eliminated by 
suitably adjusting the zero of the time scale. Thus the equations when 
written 

y — a cos (wt — kx) 
and y = a sin (ut — kx) (He) 

will describe the wave of Fig. 1 1 C, if the curve applies at times t = T/4., 
and T/2, respectively, instead of at t = 0. 

A beam of light to which equations of the above type would apply 
has the following characteristics: Not only is it a perfectly parallel beam, 
but it is absolutely monochromatic because it possesses one accurately 
defined wavelength. It is also plane-polarized, since the vibrations occur 
in a plane passing through the direction of propagation. Particularly 
as regards its monochromatic character, such light is idealized, and 
impossible of actual realization. One obvious reason is that an equation 
like Eq. lie sets no limit on x and requires an infinitely long train of 
waves. The light of a single, sharp spectrum line does, however, approach 
this ideal rather closely. 

11.3. Phase and Phase Difference. The important characteristic of 
plane waves is that the vibratory motion of every part of the medium 
is the same except for its phase. This term refers to the quantity in 
parentheses in Eq. lie, namely, the argument of the sine or cosine, and 
tells us what fraction of a complete vibration the particle has executed 
at a given instant. In one vibration, the phase increases by 2ir. Giving 
t a particular value in the equations, we see that the phase varies along 

* The physical meaning of k is that it represents the number of waves in a distance 
of 2jt cm. Hence it is sometimes called the wave number. We shall reserve this 
term for the number of waves in 1 cm. See Sec. 14.14, where this quantity is desig- 
nated by a. 






196 PHYSICAL OPTICS 

the wave in direct proportion to x. The proportionality constant is the 
propagation number k, usually expressed in radians per centimeter. The 
phase difference at any instant between two particles at positions xi and x x 
is therefore 

8 = k{xz — Xi) = — A 

A 

Here we have represented by A the difference in the x coordinate of the 
two particles, which, for reasons to be explained shortly, we call the 
path difference. 

Only differences of phase are important. The absolute value of a 
particular phase cannot be measured for light, and need never concern us. 
Differences, on the other hand, may be determined to a high degree of 
precision, and there is no arbitrariness in their definition. Similarly, the 
instantaneous displacement y is of little significance, since it is specified 
by the amplitude and absolute phase. Amplitudes and phase differences 
are the essential quantities, as will become clear in the chapters immedi- 
ately following. 

An example of the kind of optical experiment where phase differences 
play an important part is as follows: a beam of monochromatic light is 
divided into two beams, by partial reflection or otherwise. The two 
are then sent over different paths and afterward recombined. The 
intensity resulting from this superposition will depend greatly on the 
exact phase difference between the two sets of waves. This difference, 
in turn, is determined by the distances traversed by the two beams in 
reaching the point of observation. The use of the term path differences 
for the quantity A indicates that it is usually a difference for two separate 
waves, not for two points in a single wave, that is of interest. In an 
experiment of this type it may be that one or more segments of the paths 
are in a substance for which the velocity of light is appreciably different 
from that in vacuum or air. In computing phase differences, one then 
uses not the actual geometrical path through such a segment, but the 
optical path [d] (Sec. 1.5), which is the product of the distance and the 
refractive index n. The necessity for this follows from the fact that the 
velocity of light waves is less in the denser medium by the factor 1/n. 
Hence, if one requires the equivalent path in vacuum, or the path the 
light would traverse in vacuum in the same time, he uses the optical 
path instead of the geometrical one. The following important relations 
then apply : 

Phase difference 5 = — X (optical path difference) 
\ 



LIGHT WAVES 



197 



Here the two sums represent the total optical paths of the two light beams 
mentioned above. 

11.4. Phase Velocity or Wave Velocity. It is now possible to state 
more precisely what actually moves with a wave. The discussion given 
in connection with Fig. 11 A may be summed up by saying that a wave 
constitutes the progression of a condition of constant phase. This 
condition might be, for instance, the crest of the wave, where the phase 
is such as to yield the maximum upward displacement. The speed with 
which a crest moves along is usually called the wave velocity, although 
the more specific term phase velocity is sometimes used. That it is 
identical with the quantity v in our previous equations is shown by 
evaluating the rate of change of the x coordinate under the condition 
that the phase remain constant. Using the form of the phase in Eq. lie, 
the latter requirement becomes 



and the wave velocity 



ut — kx = const. 



_ dx _ oi 
v ~ ~dl~ k 



(110) 



Substitution of w = 2wv and A: = 2ir/X gives agreement with Eq. lid. 
For a wave traveling toward —x, the constant phase takes the form 
at + kx, and the corresponding v = — u/k. 

The ratio w/k for a given kind of waves depends on the physical 
properties of the medium in which the waves travel and also, in general, 
on the frequency co itself. For transverse elastic waves involving dis- 
tortions small enough so that the forces obey Hooke's law, the wave 
velocity is independent of frequency 



and is given simply as 



» = * — 



(11*) 



1 



l*X+$Z 




Fig. 11D. Illustrating the shear caused 
by a transverse wave. 



N being the shear modulus and p the 

density. The proof of this relation 

is not difficult. From Fig. 11D it 

will be seen that the sheet of small 

thickness 8x is sheared through the 

angle a. The shear modulus is the 

constant ratio of stress to strain. The strain is measured by tan a, so 

that 

Strain = -^ 
8x 

where /is the function giving the shape of the wave at a particular instant. 
The stress is the tangential force F per unit area acting on the surface 



198 PHYSICAL OPTICS 

of the sheet, and this by Hooke's law must equal the product of the shear 
modulus and the strain, so that 

Stress = F x = N^ 
8x 

Because of the curvature of the wave, the stress will vary with x, and the 
force acting on the left side of the sheet will not be exactly balanced 
by the force acting on its right side. The net force per unit area is 



Tx 8x = N d 



F x - F x+ix = -8x = N-^8x 



We now apply Newton's second law of motion, equating this force to the 
product of the mass and the acceleration of unit area of the sheet. > 

N ax-> bx = pbx w 

Comparison with the wave equation 116 then verifies the expression 
for the velocity given in Eq. \\h. 

From the fact that they can be polarized (Chap. 24), light waves 
are known to be transverse waves. Measurements show that their 
velocity in vacuum is approximately 3 X 10 10 cm/sec. If one assumes 
then to be elastic waves, as was commonly done in the nineteenth century, 
the question arises as to the medium that transmits them. Since the 
velocity is so large, Eq. \\h would require that the ratio of rigidity to 
density would have to be very great. In the early elastic-solid theory, a 
medium called the "aether" having this property was assumed to occupy 
all space. Its density was supposed to increase in material substances 
to account for the lower velocity. There are obvious objections to such 
a hypothesis. For example, in spite of its resistance to shear, which 
had to be postulated because light waves are transverse, the aether 
produces no detectable effects on the motions of astronomical bodies. 
All the difficulties disappeared when Maxwell developed the present 
electromagnetic theory of light (Chap. 20). Here the mechanical displace- 
ment of an element of the medium is replaced by a variat ion of the electric 
field (or more generally of the dielectric displacement) at the correspond- 
ing point. 

The elastic-solid theory was successful in explaining a number of 
properties of light. There are many parallelisms in the two theories, 
and much of the mathematics of the earlier theory can be rewritten in 
electromagnetic terms without difficulty. Consequently, we shall fre- 
quently find mechanical analogies useful in understanding the behavior 
of light. In fact, for the material presented in the next seven chapters, 
it is immaterial what type of waves are assumed. 



LIGHT WAVES 199 

11.5. Amplitude and Intensity. Waves transport energy, and the 
amount of it that flows per second across unit area perpendicular to the 
direction of travel is called the intensity of the wave. If the wave flows 
continuously with the velocity v, there is a definite energy density, or total 
energy per unit volume. All the energy contained in a column of the 
medium of unit cross section and of length v will pass through the unit 
of area in 1 sec. Thus the intensity is given by the product of v and the 
energy density. Either the energy density or the intensity is proportional 
to the square of the amplitude and to the square of the frequency. To 
prove this proposition for sine waves in an elastic medium, it is necessary 
only to determine the vibrational energy of a single particle executing 
simple harmonic motion. 

Consider for example the particle P in Fig. 11C. At the time for 
which the figure is drawn, it is moving upward and possesses both kinetic 
and potential energy. A little later it will have the position P'. Here 
it is instantaneously at rest, with zero kinetic energy and the maximum 
potential energy. As it subsequently moves downward, it gains kinetic 
energy, while the potential energy decreases in such a way that the total 
energy stays constant. When it reaches the center, at P", the energy 
is all kinetic. Hence we may find the total energy either from the 
maximum potential energy at P' or from the maximum kinetic energy 
at P" . The latter procedure gives the desired result most easily. 

According to Eq. lie, the displacement of a particular particle varies 
with time according to the relation 

y = a sin (cot — a) 

where a is the value of kx for that particle. The velocity of the particle is 

dy r a \ 

—r = tea cos {cot — a) 

When y = 0, the sine vanishes and the cosine has its maximum value. 
Then the velocity becomes — coa, and the maximum kinetic energy 



i-BL-i-"" 



Since this is also the total energy of the particle and is proportional to 
the energy per unit volume, it follows that 

Energy density ~ co 2 a 2 (lit) 

The intensity, v times this quantity, will then also be proportional to 
to 2 and a 2 . 

In spherical waves, the intensity decreases as the inverse square of 
the distance from the source. This law follows directly from the fact 



200 PHYSICAL OPTICS 

that, provided there is no conversion of the energy into other forms, the 
same amount must pass through any sphere with the source as its center. 
Since the area of a sphere increases as the square of its radius, the energy 
per unit area at a distance r from the source, or the intensity, will vary 
as 1/r 2 . The amplitude must then vary as 1/r, and one may write the 
equation of a spherical wave as 

y = - sin (o>t — kr) (llj) 

Here a means the amplitude at unit distance from the source. 

If any of the energy is transformed to heat, that is to say, if there is 
absorption, the amplitude and intensity of plane waves will not be con- 
stant but will decrease as the wave passes through the medium. Similarly 
with spherical waves, the loss of intensity will be more rapid than is 
required by the inverse-square law. For plane waves, the fraction dl/I 
of the intensity lost in traversing an infinitesimal thickness dx is pro- 
portional to dx, so that 

dl , 

—j- = —a ax 

To obtain the decrease in traversing a finite thickness x, the equation is 
integrated to give 



/ —f-=—a\ dx 
Jo 1 Jo 



Evaluating these definite integrals, we find 

h = he"" (11/fc) 

This law, which has been attributed to both Bouguer* and Lambert, f 
we shall refer to as the exponential law of absorption. Figure HE is a 
plot of the intensity against thickness according to this law for a medium 
having a = 0.4 per cm. The wave equations may be modified to take 
account of absorption by multiplying the amplitude by the factor e~ axn , 
since the amplitude varies with the square root of the intensity. 

For light, the intensity can be expressed in ergs per square centimeter 
per second. Full sunlight, for example, has an intensity in these units 
of about 1.4 X 10 6 . Here it is important to realize that not all this 
energy flux affects the eye, and not all that does is equally efficient. 
Hence the intensity as defined above does not necessarily correspond to 

* Pierre Bouguer (1698-1758). Royal Professor of Hydrography at Le Havre. 

t Johann Lambert (1728-1777). German physicist, astronomer, and mathe- 
matician. Worked primarily in the field of heat radiation. Another law, which is 
always called Lambert's law, refers to the variation with angle of the radiation from 
a surface. 



LIGHT WAVES 



201 



the sensation of brightness, and it is more usual to find light flux expressed 
in visual units.* The intensity and the amplitude are the purely phys- 
ical quantities, however, and according to modern theory the latter must 
be expressed in electrical units. Thus it may be shown that according 
to equations to be derived in Chap. 20 the amplitude in a beam of sun- 
light having the above-mentioned value of the intensity represents an 
electric field strength of 7.3 volts/cm and an accompanying magnetic 
field of 0.024 gauss. 




"0 1 | 2 3 f 4 5 

1.74 3.48 

Fig. HE. Decrease of intensity in an absorbing medium. 

The amplitude of light always decreases more or less rapidly with 
the distance traversed. Only for plane waves traveling in vacuum, such 
as the light from a star coming through outer space, is it nearly constant. 
The inverse-square law of intensities may be assumed to hold for a small 
source in air at distances greater than about ten times the lateral dimen- 
sion of the source. Then the finite size of the source causes an error of 
less than one-tenth of 1 per cent in computing the intensity, and for 
laboratory distances the absorption of air may be neglected. In greater 
thicknesses, however, all "transparent" substances absorb an appreciable 
fraction of the energy. We shall take up this subject again in some detail 
in Chap. 22. 

11.6. Frequency and Wavelength. Any wave motion is generated by 
some sort of vibrating source, and the frequency of the waves is equal 
to that of the source. The wavelength in a given medium is then deter- 
mined by the velocity in that medium and by Eq. lid is obtained by 
dividing the velocity by the frequency. Passage from one medium to 
another causes a change in the wavelength in the same proportion as it 
does in the velocity, since the frequency is not altered. If we remember 
that a wave front represents a surface on which the phase of motion is 

* See, for example, F. W. Sears, "Principles of Physics," vol. 3, Optics, 3d ed., 
chap. 13, Addison- Wesley Publishing Company, Cambridge, Mass., 1948. 



202 PHYSICAL OPTICS 

constant, it should be clear that, regardless of any changes of velocity, 
two different wave fronts are separated by a certain number of waves. 
That is, the length of any ray between two such surfaces is the same, 
provided this length is expressed in wavelengths in the appropriate 
media. 

As applied to light, the last statement is equivalent to saying that 
the optical path is the same along all rays drawn between two wave 
fronts. For since wavelengths are proportional to velocities, we have 

X c 
r- = - = n 

Xm V 

when the light passes from a vacuum, where it has wavelength X and 
velocity c, into a medium where the corresponding quantities are X m 
and v. The optical path corresponding to a distance d in any medium 
is therefore 

nd = — d 

X m 

or the number of wavelengths in that distance multiplied by the wave- 
length in vacuum. It is customary in optics and spectroscopy to refer 
to the "wavelength" of a particular radiation, of a single spectral line, 
for example, as its wavelength in air under normal conditions. This 
we shall designate by X (without subscript), and except in rare circum- 
stances it may be taken as the same as the wavelength in vacuum. 

The wavelengths of visible light extend between about 4 X 10 -6 cm 
for the extreme violet and 7.2 X 10 -5 cm for the deep red. Just as the 
ear becomes insensitive to sound above a certain frequency, so the eye 
fails to respond to light vibrations of frequencies greater than that of 
the extreme violet or less than that of the extreme red. The limits, of 
course, depend somewhat upon the individual, and there is evidence that 
most persons can see an image with light of wavelength as short as 
3.0 X 10 -6 cm, but this is a case of fluorescence in the retina. In this 
case the light appears to be bluish gray in color and is harmful to the 
eye. Radiation of wavelength shorter than that of the visible is termed 
"ultraviolet light" down to a wavelength of about 5 X 10 -7 cm, and 
beyond this we are in the region of X rays to 6 X 10 -10 cm. Shorter 
than these, in turn, are the 7 rays from radioactive substances. On the 
long- wavelength side of the visible lies the infrared, which may be said 
to merge into the radio waves at about 4 X 10 -2 cm. Figure IIF shows 
the names which have been given to the various regions of the spectrum 
of radiation, though we know that no real lines of demarcation exist. It 
is not convenient to use the same units of length throughout such an 
enormous range. Hence radio wavelengths are expressed in meters 



LIGHT WAVES 



203 



(10 2 cm), infrared in microns (1 n = 10 -4 cm), visible and ultraviolet, in 
angstrom* units (1 A = 10 -8 cm) and X rays in angstroms, or, com- 
monly in accurate work, in X units (1 XU = 10~ u cm). 

It will be seen that visible light covers an almost insignificant fraction 
of this range. Therefore, although all these radiations are similar in 
nature, differing only in wavelength, the term "light" is conventionally 
extended only to the adjacent portions of the spectrum, namely, the 
ultraviolet and infrared. Many of the results that we shall discuss for 
light are common to the whole range of radiation, but naturally there 



ELECTROMAGNETIC WAVES 



i ? 2 s l 

! ' A T E 
jOD T j| 



GAMMA 
RAYS 



X-RAYS 



fvT 

ULTRA jsj 
VIOLET JBj 

Lii. 



INFRA-RED 



SHORT RADIO 
WAVES 



I I 



10"" 10" 10 10" 9 10" 8 10" 7 10" 6 10" 5 10" 4 10" 3 10~ 2 Kf 1 1 10 10 2 10 3 



1XU 10XU 10 2 XU 10 3 XU 10 4 XU 



\H 10(1 I0 2 p 10 3 »i 10 4 /i 10 % 10 6 /i 



1A 10A KTA 10 3 A lO'A 



lm 10m 10 meters 



Fig. llf. Scale of wavelengths for the range of known electromagnetic waves. 



are qualitative differences in behavior between the very long and very 
short waves, which we shall occasionally point out. The divisions 
between the different types of radiation are purely formal and are roughly 
fixed by the fact that in the laboratory the different types are generated 
and detected in different ways. Thus the infrared is emitted copiously 
by hot bodies, and is detected by an energy-measuring instrument such 
as the thermopile. The shortest radio waves are generated by electric 
discharges between fine metallic particles immersed in oil and are detected 
by electrical devices. Nichols and Tear, in 1917, produced infrared 
waves having wavelengths up to 0.42 mm and radio waves down to 0.22 
mm. The two regions may therefore be said to overlap, keeping in mind, 
however, that the waves themselves are of the same nature for both. 
The same holds true for the boundaries of all the other regions of the 
spectrum. 

In sound and other mechanical waves, a change of wavelength occurs 
when the source has a translational motion. The waves sent out in 
the direction of motion are shortened, and, in the opposite direction, 
lengthened. No change is produced in the velocity of the waves them- 

* A. J. Angstrom (1814-1874). Professor of physics at Uppsala, Sweden. Chiefly 
known for his famous atlas of the solar spectrum, which was used for many years as 
the standard for wavelength determinations. 



204 PHY8ICAL OPTICS 

selves; so a stationary observer receives a frequency which is larger or 
smaller than that of the source. If, on the other hand, the source is at 
rest and the observer in motion, a change of frequency is also observed, 
but for a different reason. Here there is no change of wavelength, but the 
frequency is altered by the change in relative velocity of the waves with 
respect to the observer. The two cases involve approximately the same 
change of frequency for the same speed of motion, provided this is small 
compared with the velocity of the waves. These phenomena are known 
as the Doppler effect* and are most commonly experienced in sound 
as changes in the acoustic pitch. 

Doppler mistakenly attributed the different colors of stars to their 
motions toward or away from the earth. Because the velocity of light 
is so large, an appreciable change in color would require that a star 
have a component of velocity in the line of sight impossibly large com- 
pared with the measured velocities at right angles to it. For most stars, 
the latter usually range between 10 and 30 km/sec, with a few as high 
as 300 km/sec. Since light travels at nearly 300,000 km/sec, the expected 
shifts of frequency are small. Furthermore, it makes little difference 
whether one assumes that the observer or the source is in motion. Sup- 
pose that the earth were moving with a velocity u directly toward a fixed 
star. An observer would then receive u/\ waves in addition to the 
number v = c/X that would reach him if he were at rest. The apparent 
frequency would be 

With the velocities mentioned, this would differ from the true frequency 
by less than 1 part in 1000. A good spectroscope can, however, easily 
detect and permit the measurement of such a shift as a displacement 
of the spectrum lines. In fact, this legitimate application of Doppler's 
principle has become a powerful method of studying the radial veloci- 
ties of stars. Figure 11G shows an example where the spectrum of /x 
Cassiopeiae, in the center strip, is compared with the lines of iron from a 
laboratory source, photographed above and below. All the iron lines 
also appear in the stellar spectrum as white lines (absorption lines) but 
are shifted toward the left, i.e., toward shorter wavelengths. Measure- 
ment shows that the increase of frequency corresponds to a velocity of 
approach of 115 km/sec, which is unusually high for stars in our own 
galaxy. The spectra of other galaxies (spiral nebulae) all show displace- 

* J. C. Doppler (1803-1853). Native of Salzburg, Austria. At the age of thirty- 
two, unable to secure a position, he was about to emigrate to America. However, at 
that time he was made professor of mathematics at the Realschule in Prague and 
later became professor of experimental physics at the University of Vienna. 



LIGHT WAVES 



205 



ments toward the red, which for the most distant ones amount to several 
hundreds of angstrom units. Such values would indicate recessional 
velocities of tens of thousands of kilometers per second, and have been 
so interpreted. It is rather interesting that here there is enough redden- 
ing to change the color of the object, as postulated by Doppler, but in 
this case it occurs for objects far too faint to be seen by the naked eye. 
In the laboratory, there have been found two ways of achieving veloci- 
ties sufficient to produce detectable Doppler shifts. By reflecting light 



Fig. 11G. Doppler shift of spectrum lines in a star. Both spectra are negatives. 
{After McKellar.) 

from mirrors mounted on the rim of a wheel rotating at high speed, one 
may produce speeds of a virtual source as high as 400 m/sec. Much 
larger values are attained by beams of atoms moving in vacuum, as will 
be discussed later in Sec. 19.17. There, it is also shown that with the 
abandonment of the material aether necessitated by relativity theory 
the distinction between the cases of source in motion and of observer 
in motion disappears. Relativity leads to an equation which is sub- 
stantially Eq. Ill, with u representing the relative velocity of approach 
or recession. 

11.7. Wave Packets. As was mentioned at the end of Sec. 11.2, no 
source of waves vibrates indefinitely, as would be required for it to pro- 
duce a true sine wave. More 
commonly the vibrations die out 
because of the dissipation of 
energy or are interrupted in some 
way. Then a group of waves of 
finite length, such as that illus- 
trated in Fig. \\H, is produced. The mathematical representation of a 
wave packet of this type is rather more complex and will be briefly discussed 
in the next chapter. Since wave packets are of frequent occurrence, how- 




N waves 
Fig. Hi/. Example of a wave packet. 



206 PHYSICAL OPTICS 

ever, some features of their behavior should be mentioned here. In the first 
place, the wavelength is not well denned. If the packet be sent through 
any device for measuring wavelengths (as, for example, light through a 
diffraction grating), it will be found to yield a continuous spread over a 
certain range AX. The maximum intensity will occur at the value of X 
indicated in Fig. IIH, but energy will appear in other wavelengths, the 
intensity dying off more or less rapidly on either side of X . The larger 
the number N of waves in the group, the smaller will be the spread AX, 
and in fact theory shows* that AX/X is approximately equal to 1/N. 
Hence only when N is very large may we consider the wave to have an 
accurately defined wavelength. 

If the medium through which the packet travels is such that the 
velocity depends on frequency, two further phenomena will be observed. 
The individual wave crests will travel with a velocity different from that 
of the packet as a whole, and the packet will spread out as it progresses. 
We then have two velocities, the wave (or phase) velocity and the group 
velocity. The relation between these will be derived in Sec. 12.7. 

In light sources, the radiating atoms emit wave trains of finite length. 
Usually, because of collisions or damping arising from other causes, these 
packets are very short. According to the theorem mentioned above, the 
consequence is that the spectrum lines will not be very narrow but will 
have an appreciable width AX. A measurement of this width will yield 
the effective "lifetime" of the electromagnetic oscillators in the atoms 
and the average length of the wave packets. A low-pressure discharge 
through the vapor of mercury containing the single isotope Hg 198 yields 
very sharp spectral lines, of width about 0.005 A. Taking the wavelength 
of one of the brightest lines, 5461 A, we may estimate that there are 
roughly 10 6 waves in a packet and that the packets themselves are some 
50 cm long. 

11.8. Reflection and Refraction. When waves are incident on a bound- 
ary between two media in which the velocity is appreciably different, the 
incident wave train is divided into reflection and refracted (or trans- 
mitted) trains. The reflected energy will be relatively greater the larger 
the change in velocity. Furthermore, transverse elastic waves will be 
partly converted into longitudinal waves at such a boundary. The fact 
that the latter are not observed for light constituted another serious 
objection to the elastic-solid theory. The refracted waves are purely 
transverse and contain all of the energy that is not reflected. In general, 
both the reflected and refracted waves travel in directions different from 
that of the incident wave. 

The relations between the latter directions agree, of course, with the 

* This theorem is proved, for example, in J. A. Stratton, " Elect romagnetic Theory," 
p. 292, McGraw-Hill Book Company, Inc., New York, 1941. 



LIGHT WAVES 



207 



behavior of light rays stated in Chap. 1, since a ray represents the direc- 
tion of flow of the energy of the waves, and this is usually perpendicular 
to the wave front (for an exception, see Sec. 26.2). The laws of reflection 
and refraction were deduced in Sec. 1.6 from Fermat's principle, but it is 
well known that they also follow from the application of Huygens' con- 
struction to the reflection and refraction of a plane wave.* In Fig. 
117(a), a ray incident on a plane surface of water is indicated by a, while 
the reflected and refracted rays are indicated by ar and at, respectively. 
A question of particular interest from the standpoint of physical optics 
is that of a possible abrupt change of phase of waves when they are reflected 





(a) (6) 

Fig. 11/. Stokes' treatment of reflection. 

from a boundary. For a given boundary the result will differ, as we 
shall now show, according to whether the waves approach from the 
side of higher velocity or from that of lower velocity. Thus, let the 
symbol a in the left-hand part of Fig. 11/ represent the amplitude (not the 
intensity) of a set of waves striking the surface, let r be the fraction of the 
amplitude reflected, and let t be the fraction transmitted. The ampli- 
tudes of the two sets of waves will then be ar and at, as shown. Now, 
following a treatment given by Stokes, f imagine the two sets reversed 
in direction, as in part (6) of the figure. Provided there is no dissipation 
of energy by absorption, a wave motion is a strictly reversible phenom- 
enon. It must conform to the law of mechanics known as the principle 
of reversibility, according to which the result of an instantaneous reversal 
of all the velocities in a dynamical system is to cause the system to 
retrace its whole previous motion. That the paths of light rays are in 
conformity with this principle has already been stated in Sec. 1.4. The 

* See, for example, J. K. Robertson, "Introduction to Physical Optics," 3d ed., 
pp. 60-67, D. Van Nostrand Company, Inc., New York, 1941. 

fSir George Stokes (1819-1903). Versatile Englishman of Pembroke College, 
Cambridge, and pioneer in the study of the interaction of light with matter. He is 
known for his laws of fluorescence (Sec. 22.6) and of the rate of fall of spheres in viscous 
fluids. The treatment referred to here was given in his "Mathematical and Physical 
Papers," vol. 2, pp. 89ff., especially p. 91. 



208 PHYSICAL OPTICS 

two reversed trains, of amplitude ar and at, should accordingly have as 
their net effect after striking the surface a wave in air equal in amplitude 
to the incident wave in part (a) but traveling in the opposite direction. 
The wave of amplitude ar gives a reflected wave of amplitude arr and a 
refracted wave of amplitude art. If we call r' and t' the fractions of the 
amplitude reflected and refracted when the reversed wave at strikes the 
boundary from below, this contributes amplitudes att' and atr' ., to the 
two waves, as indicated. Now, since the resultant effect must consist 
only of a wave in air of amplitude a, we have 

att' + arr = a (11m) 

and art + atr' = (Jin) 

The second equation states that the two incident waves shall produce no 
net disturbance on the water side of the boundary. From Eq. 11m we 
obtain 

W = 1 - r 2 (llo) 

and from Eq. lln 

r' = -r (lip) 

It might at first appear that Eq. llo could be carried further by using 
the fact that intensities are proportional to squares of amplitudes and by 
writing, by conservation of energy, r 2 -\- t 2 = 1. This would immediately 
yield t = t'. The result is not correct, however, for two reasons. First, 
although the proportionality of intensity with square of amplitude holds 
for light traveling in a single medium, passage into a different medium 
brings in the additional factor of the index of refraction in determining 
the intensity. Second, it is not to the intensities that the conservation 
law is to be applied, but rather to the total energies of the beams. When 
there is a change in width of the beam, as in refraction, it must also be 
taken into account. 

The second of Stokes' relations, Eq. lip, shows that the reflectance, 
or fraction of the intensity reflected, is the same for a wave incident 
from either side of the boundary, since the negative sign disappears 
upon squaring the amplitudes. It should be noted, however, that the 
waves must be incident at angles such that they correspond to angles of 
incidence and refraction. The difference in sign of the amplitudes in 
Eq. lip indicates a difference of phase of ir between the two cases, since 
a reversal of sign means a displacement in the opposite sense. If there 
is no phase change on reflection from above, there must be a phase change 
of ir on reflection from below; or correspondingly, if there is no change on 
reflection from below, there must be a change of ir on reflection from 
above. 

The principle of reversibility as applied to light waves is often useful 



LIGHT WAVES 209 

in optical problems; for example, it proves at once the interchangeability 
of object and image. The conclusion reached above about the change 
of phase is not dependent on the applicability of the principle, i.e., on 
the absence of absorption, but holds for reflection from any boundary. 
It is a matter of experimental observation that in the reflection of light 
under the above conditions, the phase change of tt occurs when the light 
strikes the boundary from the side of higher velocity, * so that the second 
of the two alternatives mentioned is the correct one in this case. A 
change of phase of the same type is encountered in the reflection of 
simple mechanical waves, such as transverse waves in a rope. Reflection 
with change of phase where the velocity decreases in crossing the bound- 
ary corresponds to the reflection of waves from a fixed end of a rope. 
Here the elastic reaction of the fixed end of the rope immediately produces 
a reflected train of opposite phase traveling back along the rope. The 
case where the velocity increases in crossing the boundary has its parallel 
in reflection from a free end of a rope. The end of the rope undergoes 
a displacement of twice the amount it would have if the rope were con- 
tinuous, and it immediately starts a wave in the reverse direction having 
the same phase as the incident wave. We shall make use of the con- 
clusions embodied in Eqs. llo and lip in discussing the interference of 
light (Sec. 14.1) and shall return to the question of the phase relations 
for reflection at any angle of incidence in Chap. 25. 

PROBLEMS 

1. Using the relations of Eq. lid, show that the phase of a sine wave may be 
variously expressed as 

t('-!) *(?-!) - *-(<-!) 

2. Plot a sine wave having v = 20 cm/sec, X = 15 cm, and a = 5 cm, as a function 
of x at time t = 0. Assume that the particle at the origin has its extreme positive 
displacement at this time. 

Ans. Sine curve of amplitude 5, having zeros at x = 3.75, 11.25, 18.75, 26.25, . . . 
cm. 

3. In the wave of Prob. 2, plot the motion of a particle at x = 78 cm as a function 
of time, from t = to t = 5 sec. 

4. A wave is expressed by y = 10 sin (6i — 0.5x), where the time is in seconds 
and the distances in centimeters. Find the velocity and acceleration of a particle 3 cm 
from the origin at time t = 24. Ans. —25.72 cm/sec; 325.2 cm/sec 2 . 

6. What will be the phase difference in radians between two particles 90 cm apart, 
measured along the wave train represented by y = 2 sin 7x(x — 240, in which x and t 
are in centimeters and seconds, respectively? 

6. Spherical waves from a point source generate the motion y = 4.2 cos Qt mm 

* See the discussion in Sec. 13.6 in connection with Lloyd's mirror. 



210 PHYSICAL OPTICS 

at a distance of 3 m from the source. Write an equation for such waves and another 
describing the motion 50 cm from the source. 

Ans. y = (12,600/r) sin (6i - kr); y = 25.2 cos (6* + <*>.) 

7. A source of plane waves vibrates according to the equation y = a sin (2-n-t/T), 
where a = 0.8 cm and T = 0.023 sec. If the waves travel at the rate of 30 cm/sec, 
find (a) the equation of the wave, y = f(x,t), and (b) the equation of motion of a par- 
ticle 8 cm from the source. 

8. Plane sine waves having a wavelength of 62 cm traverse a certain medium. 
At a particular instant, one of the particles has a displacement of +2.6 mm, and this 
displacement is increasing. Find (a) the amplitude of the wave if the phase of the 
particle is 72° at that instant, counting the zero phase from the time the particle 
passes through its equilibrium position in the positive direction, (b) the displacement 
and phase of another particle 19 cm farther on. 

Ans. (a) 2.73 mm. (6) -0.015 mm; 180.32°. 

9. Find the velocity of transverse waves in bulk aluminum. 

10. One arm of a Michelson interferometer has a transparent, plane-parallel glass 
plate 5 mm thick, and of index no = 1.5360, set at exactly 45° with the light beam. 
The beam traverses the plate twice. Find the change in the optical path when the 
inclination of the plate is altered by 30 minutes of arc. Ans. 0.0298 mm. 

11. On'comparing the spectra from the east and west limbs (edges) of the sun, it is 
found that a spectrum line at 4126 A is shifted by 0.029 A from one spectrum to the 
other. What quantitative information regarding the motion of the sun can be 
derived from this observation? 

12. Calculate the ratio of the intensities of two waves represented by ;/i = 6 sin (OAt 
- 25x) and y, = 2.5 sin (3.2< - 200i). Ans. 0.09. 

13. The transmission coefficient of a substance is denned as the fraction of light 
that is transmitted by unit thickness. Derive a relation between this coefficient 
and the absorption coefficient as it occurs in the exponential law of absorption. 

14. Visible light spreads out from a point source under water that has a coefficient 
of absorption a = 0.08 per m. If the intensity is 2300 ergs/cm 2 sec at 50 cm from the 
source, what will it be 10 m away? Ans. 2.691 ergs/cm 2 sec. 

16. If a spectrum line in the infrared at X = 6.3 n is found to have a true width 
(corrected for any instrumental broadening) of 6 X 10 _1 /*, find the average number 
of waves in the wave packets and the average life of the molecular oscillators emitting 
the line. 

16. A parallel beam of light enters water (n = 1.330) at an angle of incidence of 
60°. Find the ratio of the width of the beam in water to that in air. Will this effect 
tend to make /' greater or smaller with respect to tl Ans. 1.518. Greater. 



CHAPTER 12 

THE SUPERPOSITION OF WAVES 



When two sets of waves are made to cross each other, as, for example, 
the waves created by dropping two stones simultaneously in a quiet pool, 
very interesting and complicated effects are observed. In the region of 
crossing there are places where the disturbance is practically zero, and 
others where it is greater than that which would be given by either wave 
alone. A very simple law can be used to explain these effects, which 
states that the resultant displacement of any point is merely the sum of 
the displacements due to each wave separately. This is known as the 
principle of superposition and was first clearly stated by Young* in 1802. 
The truth of this principle is at once evident when we observe that after 
the waves have passed out of the region of crossing, they appear to have 
been entirely uninfluenced by the other set of waves. Amplitude, fre- 
quency, and all other characteristics are just as if they had crossed an 
undisturbed space. This could hold only provided the principle of super- 
position were true. Two different observers can see different objects 
through the same aperture with perfect clearness, whereas the light 
reaching the two observers has crossed in going through the aperture. 
The principle is therefore applicable with great precision to light, and 
we may use it in investigating the disturbance in regions where two or 
more light waves are superimposed. 

12.1. Addition of Simple Harmonic Motions along the Same Line. 
Considering first the effect of superimposing two sine waves of the same 
frequency, the problem resolves itself into finding the resultant motion 
when a particle executes two simple harmonic motions at the same time. 
The displacements due to the two waves are here taken to be along the 
same line, which we shall call the y direction. If the amplitudes of the 
two waves are a x and a*, these will be the amplitudes of the two periodic 

* Thomas Young (1773-1829). English physician and physicist, usually called the 
founder of the wave theory of light. An extremely precocious child (he had read the 
Bible twice through at the age of four), he developed into a brilliant investigator. His 
work on interference constituted the most important contribution on light since New- 
ton. His early work proved the wave nature of light but was not taken seriously by 
others until it was corroborated by Fresnel. 

211 



212 PHYSICAL OPTICS 

motions impressed on the particle, and, according to Eq. lie of the last 
chapter, we may write the separate displacements as follows: 

Vl - ax sin {fd - «i) 1 (12a) 

y 2 = a 2 sin (cot — a 2 ) ) 

Note that co is the same for both waves, since we have assumed them to 
be of the same frequency. According to the principle of superposition, 
the resultant displacement y is merely the sum of yi and y 2 , and we have 

y = a,\ sin (cot — cti) + a 2 sin (cot — a 2 ) 

Using the expression for the sine of the difference of two angles, this may 
be written 

y = Oi sin cot cos ai — ai cos cot sin «i + a 2 sin cot cos a 2 

— a 2 cos cot sin a 2 
= (ai cos «i + a 2 cos a 2 ) sin cot — (ai sin ai + a 2 sin a 2 ) cos w£ (126) 

Now since the a's and a's are constants, we are justified in setting 
ai cos «i + ci2 cos a 2 = A cos 



ai sin ai + «2 sin a 2 = A sin 5 J 

provided that constant values of A and 6 can be found which satisfy 
these equations. Squaring and adding Eqs. 12c, we have 

A 2 (cos 2 + sin 2 6) = a x 2 (cos 2 ai + sin 2 a x ) + a 2 2 (cos 2 a 2 + sin 2 a 2 ) 

+ 2aia 2 (cos ai cos a 2 + sin ai sin a 2 ) 
or A 2 = ai 2 + a 2 2 + 2ai0 2 cos (ai - a 2 ) (I2d) 

Dividing the lower equation 12c by the upper one, we obtain 

di sin ai + «2 sin a 2 n „ v 

tan = j Cl^e) 

a t cos oi + a-2 cos a 2 

Equations 12d and 12e show that values of A and exist which satisfy 
Eqs. 12c, and we may rewrite Eq. 126, substituting the right-hand mem- 
bers of Eq. 12c. This gives 

y = A cos 6 sin cot — A sin cos cot 

which has the form of the sine of the difference of two angles and can be 
expressed as 

y = A sin (at - 0) (12/) 

This equation is the same as either of our original equations for $he 
separate simple harmonic motions but contains a new amplitude A and a 
new phase constant 0. Hence we have the important result that the 
sum of two simple harmonic motions of the same frequency and along 



THE SUPERPOSITION OF WAVES 213 

the same line is also a simple harmonic motion of the same frequency. 
The amplitude and phase constant of the resultant motion can easily be 
calculated from those of the component motions by Eqs. 12d and 12e, 
respectively. 

The addition of three or more simple harmonic motions of the same 
frequency will likewise give rise to a resultant motion of the same type, 
since the motions can be added successively, each time giving an equation 
of the form of Eq. 12/. Unless considerable accuracy is desired, it is 
usually more convenient to use the graphical method described in the 
following section. A knowledge of the resultant phase constant 8, given 
by Eq. 12e, is not of interest unless it is needed in combining the resultant 
motion with still another. 

The resultant amplitude A depends, according to Eq. 12d, upon the 
amplitudes a x and a 2 of the component motions and upon their difference 
of phase 8 = «i — a 2 . When we bring together two beams of light, as is 
done in the Michelson interferometer (Sec. 13.8), the intensity of the 
light at any point will be proportional to the square of the resultant 
amplitude. By Eq. 12d we have, in the case where a\ = a*, 

I ~ A 2 = 2a 2 (l + cos 8) = _4a 2 cos 2 1 (12a) 

If the phase difference is such that 8 = 0, 2*-, 4*-, . . . , then this gives 
4a 2 , or four times the intensity of either beam. If 5 = t, Sir, ox, . . . , 
the intensity is zero. For intermediate values, the intensity varies 
between these limits according to the square of the cosine. These 
modifications of intensity obtained by combining waves are referred to 
as interference effects, and we shall discuss in the next chapter several 
ways in which they may be brought about and used experimentally. 

12.2. Vector Addition of Amplitudes. A very simple geometrical con- 
struction can be used to find the resultant amplitude and phase constant 
of the combined motion in the above case of two simple harmonic motions 
along the same line. If we represent the amplitudes a,\ and a 2 by vectors 
making angles a\ and a 2 with the x axis,* as in Fig. \2A (a), the resultant 
amplitude A is the vector sum of a x and a 2 and makes an angle 6 with 
that axis. To prove this proposition, we first note from Fig. 12 A (6) 
that, in the triangle formed by a h a 2 , and A, the law of cosines gives 

A 2 = Oi 2 + a 2 2 — 2a x a 2 cos [ir — (a x — « 2 )] (12h) 

which readily reduces to Eq. 12d. Furthermore, Eq. 12e is obtained 
at once from the fact that the tangent of the angle 6 is the quotient of the 

* Here we depart from the usual convention of measuring positive angles in the 
counterclockwise direction, because it is customary in optics to represent an advance 
of phase by a clockwise rotation of the amplitude vector. 



214 



PHYSICAL OPTICS 



sum of the projections of a\ and a* on the y axis by the sum of their 
projections on the x axis. 

That the resultant motion is also simple harmonic can be concluded 
if we remember that this type of motion may be represented as the pro- 
jection on one of the coordinate axes of a point moving with uniform 




(a) (b) 

Fig. 12A. Graphical composition of amplitudes as vectors. 

circular motion. Figure 12A is drawn for the time t = 0, and as time 
progresses, the displacements yi and yi will be given by the vertical 
components of the vectors a x and a 2 , if the latter revolve clockwise with 

the same angular velocity co. The re- 
sultant, A, will then have the same 
angular velocity, and the projection P' 
of its terminus P will undergo the re- 
sultant motion. If one imagines the 
vector triangle in part (6) of the figure 
to revolve as a rigid frame, it will be 
seen that the motion of P' will agree 
with Eq. 12/. 

The graphical method is particularly 
useful where we have more than two 
motions to compound. Figure 12B 
shows the result of adding five motions 
of equal amplitudes a and having equal 
phase differences 8. Clearly the inten- 
sity I = A 2 can here vary between zero and 25a 2 , according to the 
phase difference 8. This is the problem which arises in finding the inten- 
sity pattern from a diffraction grating, as discussed in Chap. 17. The 
five equal amplitudes shown in the figure might be contributed by five 



V 




/\l 








/ V 






/ 






a/ 


a 


5 
/ 


a— 




is i 




Am 


Afh* rv s 



Fig. 1211. Vector addition of five 
amplitudes having the same magni- 
tude and phase differences 5. 



THE SUPERPOSITION OF WAVES 215 

apertures of a grating, an instrument which has as its primary purpose 
the introduction of an equal phase difference in the light from each 
successive pair of apertures. It will be noted that as Fig. 125 is drawn 
the vibrations, starting with that at the origin, lag successively further 
behind in phase. 

Either the trigonometric or graphical methods for compounding 
vibrations may be used to find the resultant of any number of motions 
with given amplitudes and phases. It is even possible, as we shall see, 
to apply these methods to the addition of infinitesimal vibrations, so 
that the summations become integrations. In such cases, and especially 
if the amplitudes of the individual contributions vary, it is simpler to 
use a method of adding the amplitudes as complex numbers. We shall 
take up this method in Sec. 14.8, where it first becomes necessary. 

12.3. Superposition of Two Wave Trains of the Same Frequency. 
From the preceding section we may conclude directly that the result of 
superimposing two trains of sine waves of the same frequency and 
traveling along the same line is to produce another sine wave of that 
frequency, but having a new amplitude which is determined for given 
values of a x and a 2 by the phase difference 8 between the motions imparted 
to any particle by the two waves. As an example, let us find the result- 
ant wave produced by two waves of equal frequency and amplitude 
traveling in the same direction +x, but with one a distance A ahead of 
the other. The equations of the two waves, in the form of Eq. 1 le, will be 

iji = a sin (u>t - kx) (I2i) 

y 2 = a sin [tat - k(x + A)] (12;) 

By the principle of superposition, the resultant displacement is the sum 
of the separate ones, so that 

V = V\ + y% - a(sin (wt — kx) 4- sin [ut - k(x 4- A)J} 
Applying the trigonometric formula 

sin A + sin B = 2 sin \{A + B) cos \(A - B) (12/b) 



we find 



y = 2a cos -=- sin 



CO/ 



-4+i)] 



(120 



This corresponds to a new wave of the same frequency but with the 
amplitude 2a cos (/cA/2) = 2a cos (ttA/\). When A is a small fraction 
of a wavelength, this amplitude will be nearly 2a, while if A is in the 
neighborhood of ?\, it will be practically zero. These cases are illus- 
trated in Fig. 12C, where the waves represented by Eqs. 12i and \2j 
(light curves) and 12/ (heavy curve) are plotted at the time t = 0. In 
these figures it will be noted that the algebraic sum of the ordinates of 



216 



PHYSICAL OPTICS 



the light curves at any value of x equals the ordinate of the heavy curve. 
The student may easily verify by such graphical construction the facts 
that the two amplitudes need not necessarily be equal to obtain a sine 
wave as the resultant and that the addition of any number of waves 
of the same frequency and wavelength also gives a similar result. In 
any case, the resultant wave form will have a constant amplitude, since 
the component waves and their resultant all move with the same velocity 
and maintain the same relative position. The true state of affairs may 




Fig. 12C. (a) Superposition of two wave trains almost in phase, (b) Superposition 
of two wave trains almost 180° out of phase. 



be pictured by having all the waves in Fig. 12C move toward the right 
with a given velocity. 

The formation of the so-called "standing waves" in a vibrating cord, 
giving rise to nodes and loops, is an example of the superposition of two 
wave trains of the same frequency and amplitude but traveling in opposite 
directions. A wave in a cord is reflected from the end, and the direct 
and reflected waves must be added to obtain the resultant motion of the 
cord. Two such waves may be represented by the equations 

iji = a sin (cot — kx) 
2/2 = a sin (ut + kx) 

By addition one obtains, in the same manner as for Eq. 12Z, 

y = 2a cos ( — kx) sin cot 

which represents the standing waves. For any value of x we have a 
simple harmonic motion, whose amplitude varies with x between the 
limits 2a, when kx = 0, v, 2-k, Sir, ... , and zero, when kx = t/2, 3t/2, 
5ir/2, .... The latter positions correspond to the nodes and are sep- 
arated by a distance Ax = rjk = X/2. Figure 12C may also serve to 



THE SUPERPOSITION OF WAVES 



217 



Parallel light waves 




Mirror 

Fig. 12D. Formation and detection of 
standing waves in Wiener's experiment. 



illustrate this case if one pictures the two lightly drawn waves as moving 
in opposite directions. The resultant curve, instead of moving unchanged 
toward the right, now oscillates between a straight-line position, when 
ut = ir/2, 3x/2, 57r/2, . . . , and a sine curve of amplitude 2a, when 
at = 0, t, 2ir, . . . . At the nodes, such as N and N' in the figure, the 
resultant displacement is zero at all times. 

The standing waves produced by reflecting light at normal incidence 
from a polished mirror may be observed by means of an experiment due 
to Wiener,* which is illustrated in Fig. 127). A specially prepared photo- 
graphic film only one-thirtieth of a 
wavelength thick is placed in an 
inclined position in front of the re- 
flecting surface so that it will cross 
the nodes and loops successively, as 
at A, a, B, b, C, c, D, d, . . . . 
The light will affect the plate 
only where there is an appreciable 
amount of vibration, and not at 
all at the nodes. As expected, the 
developed plate showed a system 

of dark bands, separated by lines of no blackening where it crossed the 
nodes. Decreasing the angle of inclination of the plate with the reflect- 
ing surface caused the bands to move farther apart, since a smaller 
number of nodal planes are cut in a given distance. On measuring 
these bands, an important fact was established: the standing waves 
have a node at the reflecting surface. The phase relations of the direct 
and reflected waves at this point are therefore such that they continu- 
ously annul each other. This is analogous to the reflection of the waves 
in a rope from a fixed end. Other experiments of a similar nature were 
performed by Wiener and these will be discussed more in detail in 
Sec. 25.12. 

12.4. Superposition of Many Waves with Random Phases. Suppose 
that we now consider a large number of wave trains of the same frequency 
and amplitude to be traveling in the same direction, and specify that the 
amount by which each train is ahead or behind any other is a matter of 
pure chance. From what has been said above, we can conclude that the 
resultant wave will be another sine wave of the same frequency, and it 
becomes of interest to inquire as to the amplitude and intensity of this 
wave. Let the individual amplitudes be a, and let there be n wave trains 
superimposed. The amplitude of the resultant wave will be the ampli- 
tude of motion of a particle undergoing n simple harmonic motions at 
once, each of amplitude a. If these motions were all in the same phase, 

* O. Wiener, Ann. Physik, 40, 203, 1890. 



218 



PHYSICAL OPTICS 



the resultant amplitude would be na and the intensity n 2 a 2 , or n 2 times 
that of one wave. In the case we are considering, however, the phases 
are distributed purely at random. If one were to use the graphical 
method of compounding amplitudes (Sec. 12.2), he would now obtain 
a picture like Fig. 12E. The phases a h a 2 , . . . take perfectly arbi- 
trary values between and 2ir. The intensity due to the superposition 
of such waves will now be determined by the square of the resultant A. 
To find A 2 , we must square the sum of the projections of all vectors a 

on the x axis and add the square of 
the corresponding sum for the y axis. 
The sum of the x projections is 




a(cos a x + cos a 2 

+ cos a 3 + 



+ cos a n ) 



Fig. 12E. Illustrating the resultant of 
12 amplitude vectors drawn with phases 
at random. 



When the quantity in parentheses 
is squared, we obtain terms of the 
form cos 2 «i and others of the form 
2 cos ai cos «2. When n is large, one 
might expect the latter terms to 
cancel out, because they take both 
positive and negative values. In 
any one arrangement of the vectors 
this is far from true, however, and in 
fact the sum of these cross-product terms actually increases approximately 
in proportion to their number. Thus we do not obtain a definite result 
with one given array of randomly distributed waves. In computing the 
intensity in any physical problem, we are always presented with a large 
number of such arrays, and we wish to find their average effect. In this 
case it is safe to conclude that the cross-product terms will average to 
zero, and we have only the cos 2 a. terms to consider. Similarly, for the 
y projections of the vectors one obtains sin 2 a terms, and the terms like 
2 sin ai sin a 2 cancel. Therefore we have 

/ ~ A 2 = a 2 (cos 2 ai + cos 2 a 2 + cos 2 a z + • • • + cos 2 a„) 

+ a 2 (sin 2 ai + sin 2 a 2 + sin 2 a 3 + • ■ ' + sin 2 a„) 

Now since sin 2 a k + cos 2 a k = 1, we find at once that 

/ ~ a 2 X n 

Thus the average intensity resulting from the superposition of n waves 
with random phases is just n times that due to a single wave. This 
means that the amplitude A in Fig. \2E, instead of averaging to zero 
when a large number of vectors a are repeatedly added in random direc- 
tions, must actually increase in length as n increases, being propor- 
tional to s/n. 



THE SUPERPOSITION OF WAVES 



219 



The above considerations may be used to explain why, when a large 
number of violins in an orchestra are playing the same note, interference 
between the sound waves need not be considered. Owing to the random 
condition of phases, 100 violins would give about 100 times the intensity 
due to one alone. The atoms in a sodium flame are emitting light with- 
out any systematic relation of phases, and furthermore each is shifting 
its phase many million times per second. Thus we may safely conclude 
that the observed intensity is exactly that due to one atom multiplied 
by the number of atoms. 



r^^sx 




(a) 



AAAA/\AA/WW\A 



m 



WVXAAAA 




tel 



VWVVW^ 



WIAJV 



wwwwwwww 



^ /v NaaW v/Wv W 



WWWWWWWV 



WWVWWWVWWV) 



V1/V* " *aA/W)A/^ - 



(d) (e) (/) 

Fig. 12F. Superposition of two or more waves traveling in the same direction with 
different relative frequencies, amplitudes, and phases. 

12.5. Complex Waves. The waves we have considered so far have 
been of the simple type in which the displacements at any instant are 
represented by a sine curve. As we have seen, superposition of any 
number of such waves having the same frequency, but arbitrary ampli- 
tudes and phases, still gives rise to a resultant wave of the same type. 
However, if only two waves having appreciably different frequencies 
are superimposed, the resulting wave is complex; i.e., the motion of 
one particle is no longer simple harmonic, and the wave contour is not 
a sine curve. The analytical treatment of such waves will be referred 
to in the following section, and here we shall consider only some of their 
more qualitative aspects. 

It is instructive to examine the results of adding graphically two or 
more waves traveling along the same line and having various relative 
frequencies, amplitudes, and phases. The wavelengths are determined 
by the frequencies according to the relation v\ = v, so that greater fre- 
quency means shorter wavelength, and vice versa. Figure 12F illustrates 



220 



PHYSICAL OPTICS 



the addition for a number of cases, the resultant curves in each case being 
obtained, according to the principle of superposition, by merely adding 
algebraically the displacements due to the individual waves at every 
point. Figure 12F(a) illustrates the case, mentioned in Sec. 12.3, of the 
addition of two waves of the same frequency but different amplitudes. 
The resultant amplitude depends on the phase difference, which in the 
figure is taken as zero. Other phase differences would be represented by 
shifting one of the component waves laterally with respect to the other 

and will give a smaller amplitude for 
the resultant sine wave, its smallest 
value being the difference in the am- 
plitudes of the components. In (6) 
three waves of different frequencies, 
amplitudes, and phases are added, 
giving a complex wave as the result- 
ant, which is evidently very different 
from a sine curve. In (c) and (d), 
where two waves of the same ampli- 
tude but frequencies in the ratio 2 : 1 
are added, it is seen that changing 
the phase difference may produce 
a resultant of very different form. 
If these represent sound waves, the 
eardrum would actually vibrate in 




Fig. 12(7. Mechanical and optical ar- 
rangement for illustrating the super- 
position of two waves. 



a manner represented by the resultant in each case, yet the ear mech- 
anism would respond to two frequencies and these would be heard and 
interpreted as the two original frequencies regardless of their phase 
difference. If the resultant wave forms represent visible light, the eye 
would similarly receive the sensation of a mixture of two colors, which 
would be the same regardless of the phase difference. Finally (e) shows 
the effect of adding a wave of very high frequency to one of very low 
frequency, and (/) the effect of adding two of nearly the same frequency. 
In the latter case, the resultant wave divides up into groups, which in 
sound produce the well-known phenomenon of beats. In any of the 
above cases, if the component waves all travel with the same velocity, 
the resultant wave form will evidently move with this velocity, keeping 
its contour unchanged. 

Experimental illustrations of the superposition of waves are easily 
accomplished with the apparatus shown in Fig. 12(7. Two small mirrors, 
Mi and M 2, are cemented to thin strips of spring steel which are clamped 
vertically and illuminated by a narrow beam of light. Such a beam is 
conveniently produced by the concentrated-arc lamp described in Sec. 
21.2. An image of this source S is focused on the screen by the lens L. 



THE SUPERPOSITION OF WAVES 221 

The beam is reflected in succession from the two mirrors, and if one of 
them is set vibrating, the reflected beam will vibrate up and down with 
simple harmonic motion. If now this beam on its way to the screen is 
reflected from a rotating mirror, the spot of light will trace out a sine 
wave form which will appear continuous by virtue of the persistence of 
vision. When both Mi and M 2 are set vibrating at once, the resultant 
wave form is the superposition of that produced by each separately. 
In this way all the curves of Fig. 12F may be produced by using two 
or more strips of suitable frequencies. The frequencies may be easily 
altered by changing the free length of the strips above the clamps. 

Since for visible light the frequency determines the color, complex 
waves of light are produced when beams of light of different colors are 
used. The "impure" colors which are not found in the spectrum will 
therefore have waves of a complex form. White light, which, since 
Newton's original experiments with prisms, we usually speak of as com- 
posed of a mixture of all colors, is the extreme example of the superposi- 
tion of a great number of waves having frequencies differing by only 
infinitesimal amounts. We shall discuss the resultant wave form for 
white light in the following section. It was mentioned in the preceding 
chapter that even the most monochromatic light we can produce in the 
laboratory still has a finite spread of frequencies. The question of the 
actual wave forms in such cases, and of how they may be described 
mathematically, should therefore be considered. 

12.6. Fourier Analysis. Since we may build up a wave of very complex 
form by the superposition of a number of simple waves, it is of interest 
to ask to what extent the converse process may be accomplished — that 
of decomposing a complex wave into a number of simple ones. Accord- 
ing to a theorem due to Fourier, any periodic function may be represented 
as the sum of a number (possibly infinite) of sine and cosine functions. 
By a periodic function we mean one that repeats itself exactly in suc- 
cessive equal intervals, such as the lower curve in Fig. 12F(b). The 
wave is given by an equation of the type 

y = do -f- <Zi sin cat + a 2 sin 2cat + a 3 sin Scot -+- • • • 

+ a[ cos cat + a\ cos 2co< + a' 3 cos Scot + • ■ ■ (12m) 

This is known as a Fourier series and contains, besides the constant term 
a , a series of terms having amplitudes oi, a 2 , . . . , a[, a' 2 , . . . and 
angular frequencies w, 2a>, 3&>, . . . . Therefore the resultant wave is 
regarded as built up of a number of waves whose wavelengths are as 
1 : ^ : i : t '• ' ' ' • In the case of sound, these represent the funda- 
mental note and its various harmonics. The evaluation of the amplitude 
coefficients a, for a given wave form can be carried out by a straightfor- 
ward mathematical process for some fairly simple wave forms but in 



\ 



222 PHYSICAL OPTICS 

general is a difficult matter. Usually one must have recourse to one 
of the various forms of "harmonic analyzer," a mechanical device for 
determining the amplitudes and phases of the fundamental and its 
harmonics.* 

Fourier analysis is not often of direct use in studying light waves, 
because it is impossible to observe directly the form of a light wave. For 
sound this can be done, and it is in the investigation of the quality of 
sounds that the Fourier analysis of waves has been most used. How- 
ever, it is important for us to understand the principles of the method, 
because, as we shall see, a grating or a prism essentially performs a 

(a) (c) (e) 

A 



^AAA^ 




ib) 




Fig. 12H. Distribution of amplitudes in different frequencies for various types of wave 
disturbance of finite length. 

Fourier analysis of the incident light, revealing the various component 
frequencies which it contains and which appear as spectral lines. 

Fourier analysis is not limited to waves of a periodic character. The 
upper part of Fig. 12H shows three types of waves which are not periodic, 
because, instead of repeating their contour indefinitely, the waves have 
zero displacement beyond a certain finite range. These " wave packets " 
(Sec. 11.7) cannot be represented by Fourier series, but instead Fourier 
integrals must be used, in which the component waves differ only by 
infinitesimal increments of wavelength. By suitably distributing the 
amplitudes for the various components, any arbitrary wave form may be 
expressed by such an integral. t The three lower curves in Fig. 12H repre- 
sent qualitatively the frequeacy distribution of the amplitudes which will 
produce the corresponding wave groups shown above. That is, the upper 

* For a detailed account of harmonic analyzers, see D. C. Miller, "The Science of 
Musical Sounds," The Macmillan Company, New York, 1922. A good discussion of 
Fourier analysis may be found in E. II. Barton, "Textbook of Sound," 1st ed., pp. 83ff ., 
The Macmillan Company, New York, 1908. 

t For a brief discussion of these integrals, and for further references, see J. A. Strai- 
ten, "Electromagnetic Theory," pp. 285-292, McGraw-Hill Book Company, Inc. 
New York, 1941. 



THE SUPERPOSITION OF WAVES 223 

curves represent the actual wave contour of the group, and this contour 
may be synthesized by adding up a very large (strictly, an infinite) num- 
ber of wave trains, each of frequency differing only infinitesimally from 
the next. The curves shown immediately below each group show the 
necessary amplitudes of the components of each frequency, in order that 
their superposition may produce the wave form indicated above. They 
represent the so-called Fourier transforms of the corresponding wave 
functions. 

Curve (a) shows the typical wave packet discussed before, and has 
the Fourier transform (6) corresponding to a single spectral line of finite 
width. The group shown in (c) would be produced by passing perfectly 
monochromatic light through a shutter which is opened for an extremely 
short time. It is worth remarking here that the corresponding amplitude 
distribution, shown in curve (d), is exactly that obtained for the Fraun- 
hofer diffraction by a single slit, as will be described in Sec. 15.3. Another 
interesting case, shown in curve (e), is that of a single pulse, such as the 
sound pulse sent out by a pistol shot or (better) by the discharge of a 
spark. The form of such a pulse may resemble that shown, and when a 
Fourier analysis is made, it yields the broad distribution of wavelengths 
shown in curve (/). For light, such a distribution is called a continuous 
spectrum and is obtained with sources of white light such as an incan- 
descent solid. The distribution of intensity in different wavelengths, 
which is proportional to the square of the ordinates in the curve, is 
determined by the exact shape of the pulse. This view of the nature of 
white light is one which has been emphasized by Gouy and others,* and 
raises the question as to whether Newton's experiments on refraction by 
prisms, which are usually said to prove the composite nature of white light, 
were of much significance in this respect. Since white light may be 
regarded as consisting merely of a succession of random pulses, of which 
the prism performs a Fourier analysis, the view that the colors are manu- 
factured by the prism, which was held by Newton's predecessors, may be 
regarded as equally correct. 

12.7. Group Velocity. It will be readily seen that, if all the com- 
ponent simple waves making up a group travel with the same velocity, 
the group will move with this velocity and maintain its form unchanged. 
If, however, the velocities vary with wavelength, this is no longer true, 
and the group will change its form as it progresses. This situation exists 
for water waves, and if one watches the individual waves in the group 
sent out by dropping a stone in still water, they will be found to be mov- 
ing faster than the group as a whole, dying out at the front of the group 

* The reader will find the more detailed discussion of the various representations of 
white light given in R. W. Wood, "Physical Optics" 1st or 2d ed., The Macmillan 
Company, New York, of interest in this connection. 



224 



PHYSICAL OPTICS 



and reappearing at the back. Hence in this case the group velocity is 
less than the wave velocity, a relation which always holds when the 
velocity of longer waves is greater than that of shorter ones. It is 
important to establish a relation between the group velocity and wave 
velocity, and this can easily be done by considering the groups formed by 
superimposing two waves of slightly different wavelength, such as those 
already discussed and illustrated in Fig. 12F(/). We shall suppose that 
the two waves have equal amplitudes, but that they have slightly differ- 
ent wavelengths, X and X', and slightly different velocities, v and v'. 

B A 



AAAAAAAAAAA/V V WWVVWVV^ w 






[O 



XX' 

"j^x" 



x = 



Fig. 12/. Illustrating groups and group velocity of two waves of slightly different 
wavelength and frequency. 

The primed quantities in each case will be taken as the larger. Then 
the propagation numbers and angular frequencies will also differ, such 
that k > k' and co > co' . The resultant wave will be given by the sum 

y = a sin (cot — kx) + a sin (co't — k'x) 

Again applying the trigonometric relation of Eq. I2fc, this equation 
becomes 

k - k' 



n . (co + co' k -\- k 
y = 2a sin ( — ^ — ' 



:' \ (co 
- X J cos I — 



t - 



'-) 



(12») 



In Figs. 127(a) and (6) the two waves are plotted separately, while (c) 
gives their sum, represented by this equation with t = 0. The resultant 
waves have the average wavelength of the two, but the amplitude is 
modulated to form groups. The individual waves, having the average 
of the two k's, correspond to variations of the sine factor in Eq. 12n, and 
according to Eq. llo their phase velocity is the quotient of the multipliers 
of t and x. 

co + co' CO 

II L — _ _ - — ->^ _ 

k + k' ~ k 
That is, the velocity is essentially that of either of the component waves, 



THE SUPERPOSITION OF WAVES 225 

since these velocities are very nearly the same. The envelope of modu- 
lation, indicated by the broken curves shown in Fig. 121, is given by the 
cosine factor. This has a much smaller propagation number, equal to the 
difference of the separate ones, and a correspondingly greater wavelength. 
The velocity of the groups is 

»-F=Tp~3Jfc (12o) 

Since no limit has been set on the smallness of the differences, they may 
be treated as infinitesimals and the approximate equality becomes exact. 
Then, since w = vk, we find for the relation between the group velocity u 
and the wave velocity v 

. . dv 

u = v + h Tk 

If the variable is changed to X, through k = 2w/X, one obtains the useful 
form 

«-»-XjS (12P) 

It should be emphasized that X here represents the actual wavelength 
in the medium. For light, this will not in most problems be the ordinary 
wavelength in air (see Sec. 23.7). 

Equations 12o and 12p, although derived for an especially simple 
type of group, are quite general and can be shown to hold for any group 
whatever, as, for example, the three illustrated in Fig. 12H(a), (c), and (e). 

The relation between wave and group velocities may also be derived 
in a less mathematical way by considering the motions of the two com- 
ponent wave trains in Fig. 127(a) and (6). At the instant shown, the 
crests A and A' of the two trains coincide to produce a maximum for the 
group. A little later the faster waves will have gained a distance X' — X 
on the slower ones, so that B' coincides with B, and the maximum of the 
group will have moved back a distance X. Since the difference in velocity 
of the two trains is dv, the time required for this is dX/dv. But in this 
time both wave trains have been moving to the right, the upper one 
moving a distance v dX/dv. The net displacement of the maximum 
of the group is thus v(dX/dv) — X in the time dX/dv, so that we obtain, 
for the group velocity, 

v(dX/dv) — X % dv 

dX/dv dX 

in agreement with Eq. 12p. 

A picture of the groups formed by two waves of slightly different fre- 
quency may easily be produced with the apparatus described in Sec. 



226 



PHYSICAL OPTICS 




12.5. It is merely necessary to adjust the two vibrating strips until the 
frequencies differ by only a few vibrations per second. 

The group velocity is the important one for light, since it is the only 
velocity which we can observe experimentally. We know of no means 
of following the progress of an individual wave in a group of light waves; 
instead, we are obliged to measure the rate at which a wave train of finite 
length conveys the energy, a quantity which can be observed. The wave 
and group velocities become the same in a medium having no dispersion, 
i.e., in which dv/d\ = 0, so that waves of all lengths travel with the 

same speed. This is accurately true 
for light traveling in a vacuum, so 
that there is no difference between 
group and wave velocities in this 
case. 

12.8. Graphical Relation between 
Wave and Group Velocity. There 
is a very simple geometrical construc- 
tion by which we may determine the 
group velocity from a curve of the 
wave velocity against wavelength. 
It is based upon the graphical inter- 
pretation of Eq. 12p. As an example, 
the curve of Fig. 12/ represents the variation of the wave velocity with 
X for water waves in deep water (gravity waves) and is drawn according 
to the theoretical equation v = const. X \/X. At a certain wavelength 
Xi, the waves have a velocity v, and the slope of the curve at the corre- 
sponding point P gives dv/dX. The line PR, drawn tangent to the curve 
at this point, intersects the v axis at the point R, the ordinate of which 
is the group velocity u for waves of wavelength in the neighborhood of Xi. 
This is evident from the fact that PQ equals Xi dv/dX, that is, the abscissa 
of P multiplied by the slope of PR. Hence QS, which is drawn equal 
to RO, represents the difference v — X dv/dX, and this is just the value of 
u, by Eq. 12p. In the particular example chosen here, it will be left 
as a problem for the student to prove that u = -|i> for any value of X. 
In water waves of this type, the individual waves therefore move with 
twice the velocity with which the group as a whole progresses. 

12.9. Addition of Simple Harmonic Motions at Right Angles. Con- 
sider the effect when two sine waves of the same frequency but having 
displacements in two perpendicular directions act simultaneously at a 
point. Choosing the directions as y and z, we may express the two 
component motions as follows: 



Fig. 12J. Graphical determination of 
group velocity from a wave velocity 
curve. 



y = ai sin (cot — ai) 
l = a? sin (cot — a 2 ) 



(12?) 



5=0 y 







/ 


a 2 

<*i 



THE SUPERPOSITION OF WAVES 



227 



/ 


7 


6 


/ 



f 


^ 


L 


J 



r 


\ 


\ 






«=* 



\ 






\ 



*= 5 % 



V 


\ 


\ 


J 



a= 3ff / 2 



5 = 7 % 



$=2* 



f 


> 


I 


J 



A 


1 


L 


/ 



/ 


I 



s- 9 % 



/ 


9 


c 


/ 



Fig. 12K. Composition at right angles of two simple harmonic motions of the same 
frequency but different phase. 

These are to be added, according to the principle of superposition, to find 
the path of the resultant motion. One does this by eliminating t from 
the two equations, obtaining 



"i 



= sin cot cos a\ — cos cot sin a\ 



— = sin cot cos «2 — cos cot sin a 2 

0,2 



(12r) 
(12s) 



Multiplying Eq. 12r by sin a 2 and Eq. 12s by sin ai and subtracting the 
first equation from the second, there results 

y z . 
— — sin a 2 -| sin a x = sin o>*(cos a 2 sin a x — cos <x x sin a 2 ) (12<) 

Similarly, multiplying Eq. 12r by cos a- 2 and Eq. 12s by cos a h and sub- 
tracting the second from the first, we obtain 

y z 

— cos a 2 cos «i = cos cot(cos a 2 sin «i — cos oti sin a 2 ) (12m) 

Cli tt 2 

We may now eliminate / from Eqs. 12/ and 12w by squaring and adding 
these equations. This gives 



y 



2yz 



sin- (a, - a 2 ) = -^ + — £- cos (ct x - a 2 ) 



(12*,) 



d\ a 2 - aia 2 
as the equation for the resultant path. In Fig. 12K the heavy curves 



228 



PHYSICAL OPTICS 



are graphs of this equation for various values of the phase difference 
5 = <x\ — «2- Except for the special cases where they degenerate into 
straight lines, these curves are all ellipses. The principal axes of the 
ellipse are in general inclined to the y and z axes but coincide with them 





la) (6) 

Fig. 12L. Graphical composition of motions in which y is (a) one-quarter and (b) 
three-quarters of a period ahead of z. 



when 5 = tt/2, 3tt/2, 5V/2, 
In this case 



, as can readily be seen from Eq. 12v. 



4 + 4 = i 

a^ a 2 2 



which is the equation of an ellipse with semiaxes a x and a->, coinciding with 
the y and z axes, respectively. When 5 = 0, 27r, 47r, . . . , we have 



ai 
y = — z 
y a 2 

representing a straight line passing through the origin, with a slope a\/ai. 

If 5 = 7r, 3ir, 5ir, • • ■ , 

ai 
y = - -z 
a 2 

a straight line with the same slope, but of opposite sign. 

That the two cases 5 = t/2 and 8 = 3ir/2, although giving the same 
path, are physically different may be seen by graphical constructions 
such as those of Fig. 12L. In both parts of the figure the motion in the 
y direction is in the same phase, the point having executed one-eighth 
of a vibration beyond its extreme positive displacement. The z motion 
in part (a) lacks one-eighth of a vibration of reaching this extreme position, 
while in part (6) it lacks five-eighths. Consideration of the directions of the 
individual motions, and of that of their resultant, will show that the 



THE SUPERPOSITION OF WAVES 



229 

In the two 



latter corresponds to the indications of the curved arrows, 
cases the ellipse is traversed in opposite senses. 

Light may be produced for which the form of vibration is an ellipse of 
any desired eccentricity. The so-called plane-polarized light (Chap. 24) 
approximates a sine wave lying in a plane — say the x,y plane of Fig. \2M 
— and the displacements are linear displacements in the y direction. If 




Fig. 123/. Composition of two sine waves at right angles. 

one combines a beam of this light with another consisting of plane- 
polarized waves lying in the x,z plane (dotted curve) and having a con- 
stant phase difference with the first, the resultant motion at any value of 
x will be a certain ellipse in the y,z plane. Such light is said to be ellip- 
tically polarized and may readily be produced by various means (Chap. 
27). A special case occurs when the amplitudes ax and a 2 of the two 
waves are equal and the phase difference is an odd multiple of ir/2. The 
vibration form is then a circle, and the light is said to be circularly polar- 
ized. When the direction of rotation is clockwise (5 = x/2, 5x/2, . . .) 
looking opposite to the direction in which the light is traveling, the light 
is called right circularly polarized, while if the rotation is counterclockwise 
(5 = 37r/2, 7tt/2, . . .), it is called left circularly polarized. 

The various types of motion shown in Fig. 12K may be readily demon- 
strated with the apparatus described in Sec. 12.5. For this purpose, 



230 PHYSICAL OPTICS 

the two strips are arranged to vibrate at right angles to each other, and 
the rotating mirror eliminated. Then one strip imparts a horizontal 
vibration to the spot of light, and the other a vertical vibration. When 
both are actuated simultaneously, the spot will trace out an ellipse. 
This will remain fixed if the two strips are tuned to exactly the same 
frequency. If they are only slightly detuned, the figure will progress 
through the forms corresponding to all possible values of the phase 
difference, passing in succession a sequence like that shown in Fig. 12K. 

PROBLEMS 

1. If two simple harmonic motions in the same line are added, these having at a 
given instant amplitudes of two and nine units and phases r/4 and ir, respectively, 
find (a) the resultant amplitude, and (b) the phase difference between the resultant 
and the first of the two motions. 

2. Plot the two equations z/i = 3 sin Girt and y 2 = sin (6irt — w/3), and obtain 
their resultant, y = A sin (Girl — 0) by addition of ordinates. Compute an exact 
value of A, and compare with that measured from the resultant curve. Ans. 3.60 cm. 

3. Find the equation for the resultant of the three motions y-i = sin (at — 10°), 
y» = 3 cos («/ + 100°), and y 3 = 2 sin (wt — 30°). The solution is to be made (a) 
by vector addition of amplitudes and (b) by computation. 

4. Find graphically the resultant amplitude when seven simple harmonic motions 
are added, each having the same period and amplitude, but each differing in phase 
from the next by 20°. Take the amplitude as 3 cm. For what value of the phase 
difference does the resultant first go to zero? Ans. 16.23 cm; 51°26'. 

5. Two waves having amplitudes of three and five units, and equal frequencies, 
are traveling in the same direction. If the phase difference is 3ir/8, find the resultant 
intensity relative to that of the separate waves. 

6. Calculate the energy of the vibration resulting from the addition of five simple 
harmonic motions having individual amplitudes a and phases 0°, 45°, 90°, 135°, and 
180°. Is the resultant amplitude increased or decreased when the fourth motion 
(having phase 135°) is removed? Ans. 5.83a 2 ; decreased. 

7. Compound graphically two wave trains having wavelengths in the ratio 4:3 
and equal amplitudes. 

8. Two sources vibrating according to the equations y\ = 3 sin irt and y 2 = sin vt 
send out plane waves which travel with a velocity of 150 cm/sec. Find the equation 
of motion of a particle 6 m from the first source and 4 m from the second. 

Ans. y = 2.65 sin fcrf 4- 19.1°). 

9. Standing waves are produced by the superposition of the two waves yi — 
18 sin (Sirt - 6x) and y 2 = 18 sin (3-irt + 6x). Find the amplitude of the motion at 
x = 23. 

10. Wiener's experiment is performed with red light (X = 6000 A). The photo- 
graphic film is inclined at ^° with the mirror, one end being in contact with it. Find 
the distances from this end of the first three dark bands produced. 

Ans. 0, 0.034, 0.069 mm. 

11. Four vibrations are capable of emitting waves of the same frequency, but of 
phases differing by only zero or w. Assuming that each possible combination of phases 
is equally probable (there are 16 of these), show that the average intensity is just 
four times that of one of the waves. Remember that the intensity due to each combi- 
nation is given by the square of the resultant amplitude. 



THE SUPERPOSITION OF WAVES 231 

12. Suppose that a beam of green light (X ^ 5200 A) consists of wave trains 6 cm 
long. What is the approximate spread of wavelengths, or the width of the spectrum 
line? Ans. 0.045 A. 

13. A square wave may be represented by a Fourier series of the form y = a sin kx. 
-f- (a/3) sin 3kx -+- (a/5) sin 5kx -f- • • • . By plotting the first three terms of tho 
series, and their sum, find how closely the resultant approximates a square wave. 

14. As suggested in Sec. 12.8, prove that for water waves controlled by gravitr 
the group velocity equals half the wave velocity. 

Ans. u - v - Mdv/d\) = C Vx - -|C y/\ = |C V> 

15. The velocity of rather short waves on the surface of a liquid is given by 



-V£F + ^) 



where T is the surface tension and d the density. Calculate the wave and group 
velocities of water waves at X = 1 cm, 5 cm, and 25 cm. 

16. For the type of waves described in Prob. 15, find the exact value of the wave- 
length for which the wave and group velocities become equal, and determine this 
velocity. Ans. 1.711 cm; 23.10 cm/sec. 

17. The phase velocity of waves in a certain medium is represented by v = Ci + C 2 X, 
where the C's are constants. What is the value of the group velocity? 

18. The refractive indices for carbon disulfide at 4900 and 6200 A are 1.65338 and 
1.62425, respectively. Assuming a Cauchy equation for n vs. X (Sec. 23.3), calculate 
the wave and group velocities of light in carbon disulfide at the mean wavelength, 
5550 A. Compare with Michelson's experimental results (group velocities, see 
Sec. 19.10) of 1/1.758 of the velocity in vacuum for white light and 1.4 per cent faster 
for "orange-red" light than for "greenish-blue." 

Ans. 1.83215 X 10 10 cm/sec; 1.70603 X 10 10 cm/sec. or c/1.7572; 4.83%. 

19. Two simple harmonic motions at right angles are represented by y = 2 sin 2wt 
and z = 5 sin (2irt — 5ir/4). Find the equation for the resultant path, and plot this 
path by the method indicated in Fig. 12L. Verify at least two points on this path 
by substitution in your resultant equation. 

20. How must the equation for the z motion in Prob. 19 be changed to yield an 
ellipse having its major axis coincident with z? To yield a circular motion with 
clockwise rotation? 

Ans. Phase constant changed to x/2, — x/2, 3*72, -3s-/2, etc.; same, but amplitude 
of z motion changed to 2. 



CHAPTER 13 

INTERFERENCE OF TWO BEAMS OF LIGHT 



It was stated at the beginning of the last chapter that two beams of 
light may be made to cross each other without either one producing any 
modification of the other after it passes beyond the region of crossing. 
In this sense the two beams do not interfere with each other. However, 
in the region of crossing, where both beams are acting at once, we are 
led to expect from the considerations of the preceding chapter that the 
resultant amplitude and intensity may be very different from the sum of 
those contributed by the two beams acting separately. This modifi- 
cation of intensity obtained by the superposition of two or more beams 
of light we call interference. If the resultant intensity is zero or in gen- 
eral less than we expect from the separate intensities, we have destructive 
interference, while if it is greater, we have constructive interference. The 
phenomenon in its simpler aspects is rather difficult to observe, because 
of the very short wavelength of light, and therefore was not recognized 
as such in the time prior to 1800 when the corpuscular theory of light 
was predominant. The first man successfully to demonstrate the inter- 
ference of light, and thus establish its wave character, was Thomas Young. 
In order to understand his crucial experiment performed in 1801, we must 
first consider the application to light of an important principle which holds 
for any type of wave motion. 

13.1. Huy gens' Principle. When waves pass through an aperture, or 
past the edge of an obstacle, they always spread to some extent into the 
region which is not directly exposed to the oncoming waves. This phe- 
nomenon is called diffraction. In order to explain this bending of light, 
Huygens nearly three centuries ago proposed the rule that each ■point on 
a wave front may be regarded as a new source of waves* This principle 
has very far-reaching applications and will be used later in discussing the 
diffraction of light, but we shall consider here only a very simple proof 

* The "waves" envisioned by Huygens were not continuous trains but rather a 
series of random pulses. Furthermore, he supposed the secondary waves to be effec- 
tive only at the point of tangency to their common envelope, thus denying the possi- 
bility of diffraction. The correct application of the principle was first made by 
Fresnel, more than a century later. 

232 



INTERFERENCE OF TWO BEAMS OF LIGHT 



233 



» 

» 
* 
» 




T) 



of its correctness. In Fig. 13 A let a set of plane waves approach the 
barrier AB from the left, and let the barrier contain an opening S of 
width somewhat smaller than the wavelength. At all points except S 
the waves will be either reflected or absorbed, but S will be free to pro- 
duce a disturbance behind the screen. It is found experimentally, in 
agreement with the above principle, that the waves spread out from S 
in the form of semicircles. 

Huygens' principle as shown in Fig. 13 A can be illustrated very suc- 
cessfully with water waves. An arc lamp on the floor, with a glass- 
bottomed tray or tank above it, 
will cast shadows of waves on a 
white ceiling. A vibrating strip of 
metal or a wire fastened to one 
prong of a tuning fork of low fre- 
quency will serve as a source of 
waves at one end of the tray. If 
an electrically driven tuning fork 
is used, the waves may be made 
apparently to stand still by placing 
a slotted disk on the shaft of a 
motor in front of the arc lamp. 
The disk is set rotating with the 
same frequency as the tuning fork 
to give the stroboscopic effect. The 

latter experiment can be performed for a fairly large audience and is well 
worth doing. Descriptions of diffraction experiments in light will be 
given in Chap. 15. 

If the experiment in Fig. 13A be performed with light, one would 
naturally expect, from the fact that light generally travels in straight 
lines, that merely a narrow patch of light would appear at D. However, 
if the slit is made very narrow, an appreciable broadening of this patch 
is observed, its breadth increasing as the slit is further narrowed. This 
remarkable evidence that light does not always travel in straight lines was 
mentioned at the very beginning of this book (Sec. 1.1 and Fig. 1A). 
When the screen CE is replaced by a photographic plate, a picture like the 
one shown in Fig. 13B is obtained. The light is most intense in the for- 
ward direction, but its intensity decreases slowly as the angle increases. 
If the slit is small compared with the wavelength of light, the intensity 
does not come to zero even when the angle of observation becomes 90°. 
While this brief introduction to Huygens' principle will be sufficient for 
an understanding of the interference phenomena we are to discuss, we 
shall return in Chaps. 15 and 18 to a more detailed consideration of dif- 
fraction at a single opening. 



E 

Fig. \ZA. Diffraction of waves at a small 
aperture. 






234 



PHYSICAL OPTICS 



13.2. Young's Experiment. The original experiment performed by 
Young is shown schematically in Fig. 13C. Sunlight was first allowed 
to pass through a pinhole S and then, at a considerable distance away, 
through two pinholes Si and «S 2 . The two sets of spherical waves emerg- 
ing from the two holes interfered with each other in such a way as to 
form a symmetrical pattern of varying intensity on the screen AC. Since 




Fig. 13B. Photograph of the diffraction of light from a slit of width 0.001 mm. 

A 




Fig. 13C Experimental arrangement for Young's double-slit experiment. 



this early experiment was performed, it has been found convenient to 
replace the pinholes by narrow slits and to use a source giving monochro- 
matic light, i.e., light of a single wavelength. In place of spherical wave 
fronts we now have cylindrical wave fronts, represented equally well in 
two dimensions by the same Fig. 13C. If the circular lines represent 
crests of waves, the intersections of any two lines represent the arrival 
at those points of two waves with the same phase or with phases differing 
by a multiple of 2t. Such points are therefore those of maximum dis- 
turbance or brightness. A close examination of the light on the screen 



INTERFERENCE OF TWO BEAMS OF LIGHT 



235 



will reveal evenly spaced light and dark bands or fringes, similar to those 
shown in Fig. 13 D. Such photographs are obtained by replacing the 
screen AC of Fig. 13C by a photographic plate. 

A very simple demonstration of Young's experiment can be accom- 
plished in the laboratory or lecture room by setting up a single-filament 
lamp L (Fig. 132?) at the front of the room. The straight vertical fila- 
ment S acts as the source and first slit. Double slits for each observer 
can be easily made from small photographic plates about 1 to 2 in. 





Fig. 13Z). Interference fringes produced by a double slit using the arrangement shown 
in Fig. 13C. 





F D\ 

Fig. 132?. Simple method for observing interference fringes. 



square. The slits are made in the photographic emulsion by drawing 
the point of a penknife across the plate, guided by a straight edge. The 
plates need not be developed or blackened but can be used as they are. 
The lamp is now viewed by holding the double slit D close to the eye E 
and looking at the lamp filament. If the slits are close together, e.g., 
0.2 mm apart, they give widely spaced fringes, whereas slits farther 
apart, e.g., 1 mm, give very narrow fringes. A piece of red glass F, 
placed adjacent to and above another of green glass in front of the lamp, 
will show that the red waves produce wider fringes than the green, which 
we shall see is due to their greater wavelength. 

Frequently one wishes to perform accurate experiments by using more 
nearly monochromatic light than that obtained by white light and a red 
or green glass filter. Perhaps the most convenient method is to use the 
sodium arc now available on the market, or a mercury arc plus a filter 
to isolate the green line, X5461. A suitable filter consists of a combination 



236 



PHYSICAL OPTICS 



of didymium glass, to absorb the yellow lines, and a light yellow glass, to 
absorb the blue and violet lines. 

13.3. Interference Fringes from a Double Source. We shall now 
derive an equation for the intensity at any point P on the screen (Fig. 
13F) and investigate the spacing of the interference fringes. Two waves 
arrive at P, having traversed different distances S 2 P and SiP. Hence 





Fig. 1SF. Path difference in Young's experiment. 

they are superimposed with a phase difference which, according to Eq. 11/, 
is 



A 



(13a) 



It is assumed that the waves start out from Si and S 2 in the same phase, 
because these slits were taken to be equidistant from the source slit S. 
Furthermore, the amplitudes are practically the same if, as is usually the 
case, $1 and S 2 are of equal width and very close together. The problem 
of finding the resultant intensity at P therefore reduces to that discussed 
in Sec. 12.1, where we considered the addition of two simple harmonic 
motions of the same frequency and amplitude, but of phase difference 5. 
The intensity was given by Eq. 12o as 



A 2 = 4a 2 cos 2 = 



(136) 



where a is the amplitude of the separate waves and A that of their 
resultant. 

It now remains to evaluate the phase difference in terms of the distance 
x on the screen from the central point Po, the separation d of the two 
slits, and the distance D from the slits to the screen. The corresponding 



INTERFERENCE OF TWO BEAMS OF LIGHT 237 

path difference is the distance S 2 A in Fig. 13F, where the dashed line SiA 
has been drawn to make Si and A equidistant from P. As Young's 
experiment is usually performed, D is some thousand times larger than d 
or x. Hence the angles 6 and 0' are very small and practically equal. 
Under these conditions, S1AS2 may be regarded as a right triangle, and 
the path difference becomes d sin 0' ~ d sin 0. To the same approxima- 
tion, we may set the sine of the angle equal to the tangent, so that 
sin ^ x/D. With these assumptions, we obtain 

A = d sin = d -^ (13c) 

This is the value of the path difference to be substituted in Eq. 13a 
to obtain the phase difference 5. Now Eq. 136 for the intensity has 
maximum values equal to 4a 2 whenever 8 is an integral multiple of 2t, 
and according to Eq. 13a this will occur when the path difference is an 
integral multiple of X. Hence we have 

xd 

j- = 0, X, 2X, 3X, . . . = mX 

Or X = mX-j BRIGHT FRINGES (13d) 

The minimum value of the intensity is zero, and this occurs when 5 = a-, 
3tt, 5ir, .... For these points 



xd _ X 3X 5X 
D " 2' T T 



••• =( m+ 3\ 



or x = (m + ~) X -j DARK fringes (13e) 



d 



The whole number m, which characterizes a particular bright fringe, is 
called the order of interference. Thus the fringes with m = 0, 1, 2, . . . 
are called the zero, first, second, etc., orders. 

According to these equations the distance on the screen between two 
successive fringes, which is obtained by changing m by unity in either 
Eq. 13d or Eq. 13e, is constant and equal to XD/d. Not only is this 
equality of spacing verified by measurement of an interference pattern 
such as Fig. 13Z), but one also finds by experiment that its magnitude 
is directly proportional to the slit-screen distance D, inversely proportional 
to the separation of the slits d, and directly proportional to the wavelength X. 
Knowledge of the spacing of these fringes thus gives us a direct determina- 
tion of X in terms of known quantities. 

These maxima and minima of intensity exist throughout the space 
behind the slits. A lens is not required to produce them, although 



238 PHYSICAL OPTICS 

they are usually so fine that a magnifier or eyepiece must be used to see 
them visually. Because of the approximations made in deriving Eq. 13c, 
careful measurements would show that, particularly in the region near 
the slits, the fringe spacing departs from the simple linear dependence 
required by Eq. IZd. A section of the fringe system in the plane of the 
paper of Fig. 13C, instead of consisting of a system of straight lines 
radiating from the mid-point between the slits, is actually a set of hyper- 
bolas. The hyperbola, being the curve for which the difference in the 
distance from two fixed points is constant, obviously fits the condi- 
tion for a given fringe, namely, the constancy of the path difference. 




Fig. 1307. Illustrating the composition of two waves of the same frequency and ampli- 
tude but different phase. 

Although this deviation from linearity may become important with sound 
and other waves, it is usually negligible when the wavelengths are as 
short as those of light. 

13.4. Intensity Distribution in the Fringe System. To find the inten- 
sity on the screen at points between the maxima, we may apply the vector 
method of compounding amplitudes described in Sec. 12.2 and illustrated 
for the present case in Fig. 13G. For the maxima, the angle 5 is zero, 
and the component amplitudes a,i and a^ are parallel, so that if they 
are equal, the resultant A = 2a. For the minima, ai and ai are in oppo- 
site directions, and A = 0. In general, for any value of 5, A is the 
closing side of the triangle. The value of A 2 , which measures the 
intensity, is then given by Eq. 136 and varies according to cos 2 (8/2). In 
Fig. 13H the solid curve represents a plot of the intensity against the 
phase difference. 

In concluding our discussion of these fringes, one question of funda- 
mental importance should be considered. If the two beams of light 
arrive at a point on the screen exactly out of phase, they interfere destruc- 
tively and the resultant intensity is zero. One may well ask what 
becomes of the energy of the two beams, since the law of conservation 
of energy tells us that energy cannot be destroyed. The answer to this 
question is that the energy which apparently disappears at the minima 
actually is still present at the maxima, where the intensity is greater than 



INTERFERENCE OF TWO BEAMS OF LIGHT 



239 



would be produced by the two beams acting separately. In other words, 
the energy is not destroyed but merely redistributed in the interference 
pattern. The average intensity on the screen is exactly that which would 
exist in the absence of interference. Thus, as shown in Fig. 13i/ , the, 
intensity in the interference pattern varies between 4a 2 and zero. Now 
each beam acting separately would contribute a 2 , and so without inter- 
ference we would have a uniform intensity of 2a 2 , as indicated by the 
broken line. To obtain the average intensity on the screen for n fringes, 
we note that the average value of the square of the cosine is £. This 
gives, by Eq. 13&, I ~ 2a 2 , justifying the statement made above, and 



7= 4a 2 cos 2 -§- 




& — "• -5tt -4tt -3tt -2jt -7T K 2k 3jt 4jt 5jr 6jr 7jr 
Fig. 13/7. Intensity distribution for the interference fringes from two beams. 

it shows that no violation of the law of conservation of energy is involved 
in the interference phenomenon. 

13.5. Fresnel's Biprism.* Soon after the double-slit experiment was 
performed by Young, the objection was raised that the bright fringes he 
observed were probably due to some complicated modification of the 
light by the edges of the slits and not to true interference. Thus the 
wave theory of light was still questioned. Not many years passed, 
however, before Fresnel brought forward several new experiments in 
which the interference of two beams of light was proved in a manner not 
open to the above objection. One of these, the so-called Fresnel biprism 
experiment, will be described in some detail. 

A schematic diagram of the biprism experiment is shown in Fig. 137. 
The thin double prism P refracts the light from the slit source 5 into 
two overlapping beams ac and be. If screens M and N are placed as 
shown in the figure, interference fringes are observed only in the region 
he. When the screen ae is replaced by a photographic plate, a picture 

* Augustin Fresnel (1788-1827). Most notable French contributor to the theory of 
light. Trained as an engineer, he became interested in light, and in 1814—1815 he 
rediscovered Young's principle of interference and extended it to complicated cases 
of diffraction. His mathematical investigations gave the wave theory a sound 
foundation. 



240 



PHYSICAL OPTICS 



like the upper one in Fig. 13J is obtained. The closely spaced fringe? 
in the center of the photograph are due to interference, while the wide 
fringes at the edge of the pattern are due to diffraction. These wider 
bands are produced by the vertices of the two prisms, each of which acts 




Fig. 13/. Diagram of Fresnel's biprism experiment. 



I be e 




Fig. 13J. Interference and diffraction fringes produced in the Fresnel biprism experi- 
ment. 

as a straight edge, giving a pattern which will be discussed in detail in 
Chap. 18. When the screens M and N are removed from the light path, 
the two beams will overlap over the whole region ae. The lower photo- 
graph in Fig. 13./ shows for this case the equally spaced interference 
fringes superimposed on the diffraction pattern of a wide aperture. (For 
the diffraction pattern above, without the interference fringes, see lowest 
figures in Fig. 18U). With such an experiment Fresnel was able to 
produce interference without relying upon diffraction to bring the inter- 
fering beams together. 

Just as in Young's double-slit experiment, the wavelength of light can 
be determined from measurements of the interference fringes produced 



INTERFERENCE OF TWO BEAMS OF LIGHT 241 

by the biprism. Calling B and C the distances of the source and screen, 
respectively, from the prism P, d the distance between the virtual images 
S\ and £2, and Ax the distance between the successive fringes on the 
screen, the wavelength of the light is given from Eq. 13d as 

Thus the virtual images Si and & 2 act as did the two slit sources in 
Young's experiment. 

In order to find d, the linear separation of the virtual sources, one may 
measure their angular separation 6 on a spectrometer and assume, to 
sufficient accuracy, that d = Bd. If the parallel light from the collimator 
covers both halves of the biprism, two images of the slit are produced and 
the angle between these is easily measured with the telescope. An even 
simpler measurement of this angle may be made by holding the prism 
close to one eye and viewing a round frosted light bulb. At a certain 
distance from the light the two images may be brought to the point 
where their inner edges just touch. The diameter of the bulb divided 
by the distance from the bulb to the prism then gives 6 directly. 

Fresnel biprisms are easily made from a small piece of glass, such as 
half a microscope slide, by beveling about i to ^ in. on one side. This 
requires very little grinding with ordinary abrasive materials, and polish- 
ing with rouge, since the angle required is only about 1°. 

13.6. Other Apparatus Depending on Division of the Wave Front. 
Two beams may be brought together in other ways to produce inter- 
ference. In the arrangement known as Fresnel' s mirrors, light from a 
slit is reflected in two plane mirrors slightly inclined to each other. The 
mirrors produce two virtual images of the slit, as shown in Fig. 13K. 
They act in every respect like the images formed by the biprism, and 
interference fringes are observed in the region be, where the reflected 
beams overlap. The symbols in this diagram correspond to those in 
Fig. 137, and Eq. 13/ is again applicable. It will be noted that the 
angle 20 subtended at the point of intersection M by the two sources is 
twice the angle between the mirrors. 

The Fresnel double-mirror experiment is usually performed on an 
optical bench, with the light reflected from the mirrors at nearly grazing 
angles. Two pieces of ordinary plate glass about 2 in. square make a 
very good double mirror. One plate should have an adjusting screw 
for changing the angle 0, and the other a screw for making the edges 
of the two mirrors parallel. 

An even simpler device, shown in Fig. 13L, produces interference 
between the light reflected in one long mirror and the light coming directly 
from the source without reflection. In this arrangement, known as 



242 



PHYSICAL OPTICS 



Lloyd's mirror, the quantitative relations are similar to those in the fore- 
going cases, with the slit and its virtual image constituting the double 
source. An important feature of the Lloyd's-mirror experiment lies 
in the fact that when the screen is placed in contact with the end of the 




Fig. 13iC. Geometry of Fresnel's mirrors. 




Fig. V.1L. Lloyd's mirror. 



mirror (in the position MN, Fig. 13L), the edge of the reflecting surface 
comes at the center of a dark fringe, instead of a bright one as might be 
expected. This means that one of the two beams has undergone a phase 
change of x. Since the direct beam could not change phase, this experi- 
mental observation is interpreted to mean that the reflected light has 
changed phase at reflection. Two photographs of the Lloyd's mirror 



INTERFERENCE OF TWO BEAMS OF LIGHT 



243 



fringes taken in this way are reproduced in Fig. 133/, one taken with 
visible light and the other with X rays. 

If the light from source Si in Fig. 13L is allowed to enter the end of 
the glass plate by moving the latter up, and to be internally reflected 
from the upper glass surface, fringes will again be observed in the interval 
OP, with a dark fringe at 0. This shows that there is again a phase 
change of t at reflection. As will be shown in Chap. 25, this is not in 
contradiction with the discussion of phase change given in Sec. 11.8. In 
this instance the light is incident at an angle greater than the critical 
angle for total reflection. 




(a) (b) 

Fig. 13M . Interference fringes produced with Lloyd's mirror, (a) Taken with visible 
light, X = 4358 A. (After While.) (b) Taken with X rays, X = 8.33 A. (After 
KeUstrom.) 

Lloyd's mirror is readily set up for demonstration purposes as follows: 
A carbon arc, followed by a colored glass filter and a narrow slit, serves 
as a source. A strip of ordinary plate glass 1 to 2 in. wide and a foot 
or more long makes an excellent mirror. A magnifying glass focused 
on the far end of the mirror enables one to observe the fringes shown in 
Fig. 133/ . Internal fringes can be observed by polishing the ends of 
the mirror to allow the light to enter and leave the glass, and by rough- 
ening one of the glass faces with coarse emery. 

Other ways exist* for dividing the wave front into two segments and 
subsequently recombining these at a small angle with each other. For 
example, one may cut a lens into two halves on a plane through the lens 
axis and separate the parts slightly, to form two closely adjacent real 
images of a slit. The images produced in this device, known as Billet's 
svlit lens, act like the two slits in Young's experiment. A single lens 
followed by a bi plate (two plane-parallel plates at a slight angle) will 
accomplish the same result. 

13.7. Coherent Sources. It will be noticed that the various methods 
of demonstrating interference so far discussed have one important feature 

* Good descriptions will be found in T. Preston, "Theory of Light, 5th ed., chap. 7, 
The Macmillan Company, New York, 1928. 



244 PHYSICAL OPTICS 

in common: The two interfering beams are always derived from the 
same source of light. We find by experiment that it is impossible to 
obtain interference fringes from two separate sources, such as two lamp 
filaments set side by side. This failure is caused by the fact that the 
light from any one source is not an infinite train of waves. On the 
contrary, there are sudden changes in phase occurring in very short 
intervals of time (of the order of 10~ 8 sec). This point has already been 
mentioned in Sees. 11.7 and 12.6. Thus, although interference fringes 
may exist on the screen for such a short interval, they will shift their 
position each time there is a phase change, with the result that no fringes 
at all will be seen. In Young's experiment and in Fresnel's mirrors and 
biprism, the two sources Si and *S 2 always have a point-to-point corre- 
spondence of phase, since they are both derived from the same source. 
If the phase of the light from a point in Si suddenly shifts, that of the light 
from the corresponding point in <S 2 will shift simultaneously. The result 
is that the difference in phase between any pair of points in the two 
sources always remain constant, and so the interference fringes are sta- 
tionary. It is a characteristic of any interference experiment with light 
that the sources must have this point-to-point phase relation, and sources 
that have this relation are called coherent sources. 

While special arrangements are necessary for producing coherent 
sources of light, the same is not true of microwaves, which are radio waves 
of a few centimeters wavelength. These are produced by an oscillator 
which emits a continuous wave, the phase of which remains constant 
over a time long compared with the duration of an observation. Two 
independent microwave sources of the same frequency are therefore 
coherent and may be used to demonstrate interference. Because of the 
convenient magnitude of their wavelength, microwaves may be used for 
illustrating many common optical interference and diffraction effects.* 

If in Young's experiment the source slit S (Fig. 13C) is made too wide, 
or the angle between the rays which leave it too large, the double slit no 
longer represents two coherent sources and the interference fringes dis- 
appear. This subject will be discussed in more detail at the end of 
Chap. 16, The Double Slit. 

13.8. Division of Amplitude. Michelsont Interferometer. Interfer- 
ence apparatus may be conveniently divided into two main classes, 

* The technique of such experiments is discussed by G. F. Hull, Jr., Am. J. Phys., 
17, 599, 1949. 

t A. A. Michelson (1852-1931). American physicist of great genius. He early 
became interested in the velocity of light, and began experiments while an instructor 
in physics and chemistry at the Naval Academy, from which he graduated in 1873. 
It is related that the superintendent of the Academy asked young Michelson why he 
wasted his time on such useless experiments. Years later Michelson was awarded 



/ 



INTERFERENCE OF TWO BEAMS OF LIGHT 



245 



those based on division of wave front and those based on division of ampli- 
tude. The previous examples all belong to the former class, in which 
the wave front is divided laterally into segments by mirrors or dia- 
phragms. It is also possible to divide a wave by partial reflection, the 
two resulting wave fronts maintaining the original width but having 
reduced amplitudes. The Michelson interferometer is an important 
example of this second class. Here the two beams obtained by amplitude 
division are sent in quite different directions against plane mirrors, 
whence they are brought together again to form interference fringes. 
The arrangement is shown sche- 
matically in Fig. 13.V. The main 
optical parts consist of two highly 
polished plane mirrors il/i and M2 
and two plane-parallel plates of 
glass G\ ana Gi. Sometimes the 
rear side of the plate G\ is lightly 
silvered (shown by the heavy line 
in the figure) so that the light 
coming from the source S is di- 
vided into (1) a reflected and (2) 
a transmitted beam of equal in- 
tensity. The light reflected nor- 
mally from mirror Mi passes 
through G\ a third time and 
reaches the eye as shown. The 
light reflected from the mirror Jf 2 
passes back through G 2 for the second time, is reflected from the surface 
of Gi and into the eye. The purpose of the plate G 2 , called the compen- 
sating plate, is to render the path in glass of the two rays equal. This 
is not essential for producing fringes in monochromatic light, but it is 
indispensable when white light is used (Sec. 13.11). The mirror jfcfi is 
mounted on a carriage C and can be moved along the well-machined 
waves or tracks T. This slow and accurately controlled motion is accom- 
plished by means of the screw V which is calibrated to show the exact 
distance the mirror has been moved. To obtain fringes, the mirrors M 1 
and M 2 are made exactly perpendicular to each other by means of screws 
shown on mirror Mi. 

Even when the above adjustments have been made, fringes will not 




Fia. 13A7. Diagram of the Michelson 
interferometer. 



the Nobel prize (1907) for his work on light. Much of his work on the velocity of 
light (Sec. 19.5) was done during 10 years spent at the Case Institute of Technology. 
During the latter part of his life he was professor of physics at the University of 
Chicago, where many of his famous experiments on the interference of light were done. 



246 PHYSICAL OPTICS 

be seen unless two important requirements are fulfilled. First, the light 
must originate from an extended source. A point source or a slit source, 
as used in the methods previously described, will not produce the desired 
system of fringes in this case. The reason for this will appear when we 
consider the origin of the fringes. Second, the light must in general be 
monochromatic, or nearly so. Especially is this true if the distances of 
My. and Mi from G\ are appreciably different. 

An extended source suitable for use with a Michelson interferometer 
may be obtained in any one of several ways. A sodium flame or a 
mercury arc, if large enough, may be used without the screen L shown 
in Fig. 13N. If the source is small, a ground glass screen or a lens at L 
will extend the field of view. Looking at the mirror Mi through the 
plate G\, one then sees the whole mirror filled with light. In order to 
obtain the fringes, the next step is to measure the distances of M x and 
M 2 to the back surface of G x roughly with a millimeter scale, and to move 
Mi until they are the same to within a few millimeters. The mirror M 2 
is now adjusted to be perpendicular to M i by observing the images of a 
common pin, or any sharp point, placed between the source and 0%. 
Two pairs of images will be seen, one coming from reflection at the front 
surface of Gi and the other from reflection at its back surface. When the 
tilting screws on M 2 are now turned until one pair of images falls exactly 
on the other, the interference fringes should appear. When they first 
appear, the fringes will not be clear unless the eye is focused on or near 
the back mirror M u so the observer should look constantly at this mirror 
while searching for the fringes. When they have been found, the 
adjusting screws should be turned in such a way as to continually increase 
the width of the fringes, and finally a set of concentric circular fringes will 
be obtained. Mi is then exactly perpendicular to M h if the latter is 
at an angle of 45° with G\. 

13.9. Circular Fringes. These are produced with monochromatic light 
when the mirrors are in exact adjustment and are the ones used in most 
kinds of measurement with the interferometer. Their origin may be 
understood by reference to the diagram of Fig. 130. Here the real 
mirror Mi has been replaced by its virtual image M' 2 formed by reflec- 
tion in G\. M' 2 is then parallel to Mi. Owing to the several reflections 
in the real interferometer, we may now think of the extended source 
as being at L, behind the observer, and as forming two virtual images 
Li and Li in Mi and M' 2 . These virtual sources are coherent in that the 
phases of corresponding points in the two are exactly the same at all 
instants. If d is the separation MiM' 2> the virtual sources will be sep- 
arated by 2d. When d is exactly an integral number of half wavelengths, 
i.e., the path difference 2d equal to an integral number of whole wave- 
lengths, all rays of light reflected normal to the mirrors will be in phase. 



INTERFERENCE OF TWO BEAMS OF LIGHT 



247 



Rays of light reflected at an angle, however, will in general not be in 
phase. The path difference between the two rays coming to the eye 
from corresponding points P' and P" is 2d cos 0, as shown in the figure. 
The angle is necessarily the same for the two rays when Mi is parallel 
to M'z so that the rays are parallel. Hence when the eye is focused to 
receive parallel rays (a small telescope is more satisfactory here, especially 

P'_2d P"_ 






M t Mz 




I*- 2d-*] 
Fig. 130. Formation of circular fringes in the Michelson interferometer. 

for large values of d) the rays will reinforce each other to produce maxima 
for those angles satisfying the relation 



2d cos = roX 



(130) 



Since for a given m, X, and d the angle is constant, the maxima will 
lie in the form of circles about the foot of the perpendicular from the eye 
to the mirrors. By expanding the cosine, it can be shown from Eq. ISg 
that the radii of the rings are proportional to the square roots of integers, 
as in the case of Newton's rings (Sec. 14.5). The intensity distribution 
across the rings follows Eq. 136, in which the phase difference is given by 

5 = =- 2d cos 

A 

Fringes of this kind, where parallel beams are brought to interference 
with a phase difference determined by the angle of inclination 0, are often 
referred to as fringes of equal inclination. In contrast to the type to be 
described in the next section, this type may remain visible over very 
large path differences. The eventual limitation on the path difference 
will be discussed in Sec. 13.12. 



248 



PHYSICAL OPTICS 



The upper part of Fig. 13P shows how the circular fringes look under 
different conditions. Starting with M i a few centimeters beyond M 2 , the 
fringe system will have the general appearance shown in (a) with the rings 
very closely spaced. If Mi is now moved slowly toward M 2 so that d 
is decreased, Eq. \Zg shows that a given ring, characterized by a given 
value of the order m, must decrease its radius because the product 
2d cos must remain constant. The rings therefore shrink and vanish at 
the center, a ring disappearing each time 2d decreases by X, or d by X/2. 




(/) (g) (h) (i) (i) 

Fig. 13P. Appearance of the various types of fringes observed in the Michelson 
interferometer. Upper row, circular fringes. Flower row, localized fringes. Path 
difference increases outward, in both directions, from the center. 



This follows from the fact that at the center cos = 1, so that Eq. 13<? 
becomes 

2d = mX (13/i) 

To change m by unity, d must change by X/2. Now as M x approaches 
M 2 the rings become more widely spaced, as indicated in Fig. 13P(6), 
until finally we reach a critical position where the central fringe has 
spread out to cover the whole field of view, as shown in (c). This hap- 
pens when Mi and M 2 are exactly coincident, for it is clear that under 
these conditions the path difference is zero for all angles of incidence. 
If the mirror is moved still farther, it effectively passes through M 2 , and 
new widely spaced fringes appear, growing out from the center. These 
will gradually become more closely spaced as the path difference increases, 
as indicated in (d) and (e) of the figure. 

13.10. Localized Fringes. If the mirrors M 2 and Mi are not exactly 
parallel, fringes will still be seen with monochromatic light for path 
differences not exceeding a few millimeters. In this case the space 
between the mirrors is wedge-shaped, as indicated in Fig. 13Q. The 



INTERFERENCE OF TWO BEAMS OF LIGHT 



/ 



249 



two rays* reaching the eye from a point P on the source are now no 
longer parallel, but appear to diverge from a point P' near the mirrors. 
For various positions of P on the extended source, it can be shown t /that 
the path difference between the two rays remains constant, but that the 
distance of P' from the mirrors changes. If the angle between the 
mirrors is not too small, however, the latter distance is never great, and 
hence, in order to see these fringes clearly, the eye must be focused on 
or near the rear mirror Mi. The localized fringes are practically straight 




Fig. 13Q. Diagram illustrating the formation of fringes with inclined mirrors in the 
Michelson interferometer. 



because the variation of the path difference across the field of view is 
now due primarily to the variation of the thickness of the "air film" 
between the mirrors. With a wedge-shaped film, the locus of points of 
equal thickness is a straight line parallel to the edge of the wedge. The 
fringes are not exactly straight, however, if d has an appreciable value, 
because there is also some variation of the path difference with angle. 
They are in general curved and are always convex toward the thin edge 
of the wedge. Thus, with a certain value of d, we might observe fringes 
shaped like those of Fig. 13P(g). Mi could then be in a position such as g 
of Fig. 13 Q. If the separation of the mirrors is decreased, the fringes 
will move to the left across the field, a new fringe crossing the center 
each time d changes by X/2. As we approach zero path difference, the 
fringes become straighter, until the point is reached where Mi actually 
intersects M' 2 , when they are perfectly straight, as in (h). Beyond this 

* When the term "ray" is used, here and elsewhere in discussing interference 
phenomena, it merely indicates the direction of the perpendicular to a wave front and 
is in no way to suggest an infinitesimally narrow pencil of light. 

t R. W. Ditchburn, "Light," 1st ed., pp. 132-134, Interscience Publishers, Inc., 
New York, 1953. 



250 



PHYSICAL OPTICS 



point, they begin to curve in the opposite direction, as shown in (*). 
The blank fields (/) and (,;') indicate that this type of fringe cannot be 
observed for large path differences. Because the principal variation of 
path difference results from a change of the thickness d, these fringes 
have been termed fringes of equal thickness. 

13.11. White-light Fringes. If a source of white light is used, no 
fringes will be seen at all except for a path difference so small that it 
does not exceed a few wavelengths. In observing these fringes, the 
mirrors are tilted slightly as for localized fringes, and the position of M i 
is found where it intersects M' 2 . With white light there will then be 
observed a central dark fringe, bordered on either side by 8 or 10 colored 




GR G R 

FlG. 13#. Illustrating the formation of white-light fringes with a dark fringe at the 
center. 

fringes. This position is often rather troublesome to find using white 
light only. It is best located approximately beforehand by finding the 
place where the localized fringes in monochromatic light becomes straight. 
Then a very slow motion of Mi through this region, using white light, 
will bring these fringes into view. 

The fact that only a few fringes are observed with white light is easily 
accounted for when we remember that such light contains all wavelengths 
between 4000 and 7500 A. The fringes for a given color are more widely 
spaced the greater the wavelength. Thus the fringes in different colors 
will only coincide for d = 0, as indicated in Fig. 1372. The solid curve 
represents the intensity distribution in the fringes for green light, and the 
broken curve that for red light. Clearly, only the central fringe will be 
uncolored, and the fringes of different colors will begin to separate at 
once on either side, producing various impure colors which are not the 
saturated spectral colors. After 8 or 10 fringes, so many colors are 
present at a given point that the resultant color is essentially white. 
Interference is still occurring in this region, however, because a spectro- 
scope will show a continuous spectrum with dark bands at those wave- 
lengths for which the condition for destructive interference is fulfilled. 
White-light fringes are also observed in all the other methods of producing 



INTERFERENCE OF TWO BEAMS OF LIGHT 251 

interference described above, if white light is substituted for monochro- 
matic light. They are particularly important in the Michelson interfer- 
ometer, where they may be used to locate the position of zero path 
difference, as we shall see in Sec. 13.13. 

An excellent reproduction in color of these white-light fringes is given 
in one of Michelson's books.* The fringes in three different colors are 
also shown separately, and a study of these in connection with the white- 
light fringes is instructive as showing the origin of the various impure 
colors in the latter. 

It was stated above that the central fringe in the white-light system, 
i.e., that corresponding to zero path difference, is black when observed 
in the Michelson interferometer. One would ordinarily expect this fringe 
to be white, since the two beams should be in phase with each other for 
any wavelength at this point, and in fact this is the case in the fringes 
formed with the other arrangements, such as the biprism. In the present 
case, however, it will be seen by referring to Fig. 13A7 that while ray (1) 
undergoes an internal reflection in the plate Gi, ray (2) undergoes an 
external reflection, with a consequent change of phase (Sec. 11.8). Hence 
the central fringe is black, if the back surface of Gi is unsilvered. If it is 
silvered, the conditions are different and the central fringe may be white. 

13.12. Visibility of the Fringes. There are three principal types of 
measurement that can be made with the interferometer: (1) width and 
fine structure of spectrum lines, (2) lengths or displacements in terms 
of wavelengths of light, and (3) refractive indices. As was explained in 
the preceding section, when a certain spread of wavelengths is present in 
the light source, the fringes become indistinct and eventually disappear 
as the path difference is increased. With white light they become 
invisible when d is only a few wavelengths, whereas the circular fringes 
obtained with the light of a single spectrum line may still be seen after 
the mirror has been moved several centimeters. Since no line is perfectly 
sharp, however, the different component wavelengths produce fringes of 
slightly different spacing, and hence there is a limit to the usable path 
difference even in this case. For the measurements of length to be 
described below, Michelson tested the lines from various sources and 
concluded that a certain red line in the spectrum of cadmium was the 
most satisfactory. He measured the visibility, defined as 

where /„,„ and 7 min are the intensities at the maxima and minima of the 
fringe pattern. The more slowly V decreases with increasing path 

* A. A. Michelson, "Light Waves and Their Uses," plate II, University of Chicago 
Press, Chicago, 1906. 



252 



PHYSICAL OPTICS 



difference, the sharper the line. With the red cadmium line, it dropped 
to 0.5 at a path difference of some 10 cm, or at d = 5 cm. 

With certain lines, the visibility does not decrease uniformly but 
fluctuates with more or less regularity. This behavior indicates that 
the line has a fine structure, consisting of two or more lines very close 
together. Thus it is found that with sodium light the fringes become 
alternately sharp and diffuse, as the fringes from the two D lines get in 
and out of step. The number of fringes between two successive positions 
of maximum visibility is about 1000, indicating that the wavelengths of 




Fig. 135. Limiting path difference as determined by the length cf wave packets. 

the components differ by approximately 1 part in 1000. In more com- 
plicated cases, the separation and intensities of the components could 
be determined by a Fourier analysis of the visibility curves.* Since 
this method of inferring the structure of lines has now been superseded 
by more direct methods, to be described in the following chapter, it will 
not be discussed in any detail here. 

There is an alternative way of interpreting the eventual vanishing of 
interference at large path differences, which it is instructive to consider 
at this point. In Sec. 12.6 it was indicated that a finite spread of wave- 
lengths corresponds to wave packets of limited length, this length 
decreasing as the spread becomes greater. Thus, when the two beams 
in the interferometer traverse distances that differ by more than the 
length of the individual packets, these can no longer overlap and no 
interference is possible. The situation upon complete disappearance 
of the fringes is shown schematically in Fig. 13<S. The original wave 

* A. A. Michelson, "Studies in Optics," chap. 4, University of Chicago Press, 
Chicago, 1927. 



INTERFERENCE OF TWO BEAMS OF LIGHT 253 

packet P has its amplitude divided at G i so that two similar packets are 
produced, Pi traveling to M h and P 2 to M 2 . When the beams are 
reunited, Pi lags a distance 2d behind Pi. Evidently a measurement of 
this limiting path difference gives a direct determination of the length 
of the wave packets. This interpretation of the cessation of interference 
seems at first sight to conflict with the one given above. A consideration 
of the principle of Fourier analysis shows, however, that mathematically 
the two are entirely equivalent and are merely alternative ways of repre- 
senting the same phenomenon. 

13.13. Interferometric Measurements of Length. The principal 
advantage of Michelson's form of interferometer over the earlier methods 
of producing interference lies in the fact that the two beams are here 
widely separated, and the path difference can be varied at will by moving 
the mirror or by introducing a refracting material in one of the beams. 
Corresponding to these two ways of changing the optical path, there 
are two other important applications of the interferometer. Accurate 
measurements of distance in terms of the wavelength of light will be 
discussed in this section, while interferometric determinations of refrac- 
tive indices are described in Sec. 13.15. 

When the mirror M i of Fig. 13Af is moved slowly from one position to 
another, counting the number of fringes in monochromatic light which 
cross the center of the field of view will give a measure of the distance the 
mirror has moved in terms of X, since by Eq. 13/i we have, for the position 
di corresponding to the bright fringe of order w x , 

2di = mik 
and for d 2 , giving a bright fringe of order m 2 , 

2d 2 = m 2 \ 
Subtracting these two equations, we find 

di - d 2 = (mi — m 2 ) ^ (13j) 

Hence the distance moved equals the number of fringes counted, multi- 
plied by a half wavelength. Of course, the distance measured need not 
correspond to an integral number of half wavelengths. Fractional parts 
of a whole fringe displacement can easily be estimated to one-tenth of a 
fringe, and, with care, to one-fiftieth. The latter figure then gives the 
distance to an accuracy of ttoA, or 5 X 10~ 7 cm for green light. 

A small Michelson interferometer in which a microscope is attached 
to the moving carriage carrying Mi is frequently used in the laboratory 
for measuring the wavelength of light. The microscope is focused on a 
fine glass scale, and the number of fringes, mi — m 2 , crossing the mirror 



254 



PHYSICAL OPTICS 



between two readings di and d 2 on the scale gives X, by Eq. IZj. The 
bending of a beam, or even of a brick wall, under pressure from the hand 

can be made visible and measured 
by attaching Mi directly to the 
beam or wall. 

The most important measure- 
ment made with the interferom- 
e1?§r was the comparison of the 
sian&^d meter in Paris with the 
wavelengths of intense red, green, 
and blue lines of cadmium by 
Michel son and Benoit. For rea- 
sons discussed in the last section, 
it would be impossible to count 
directly the number of fringes for a displacement of the movable mirror 
from one end of the standard meter to the other. Instead, nine inter- 
mediate standards (etalons) were used, of the form shown in Fig. 13 T, 
each approximately twice the length of the other. The two shortest 




Fig. 13 T. One of the nine etalons used by 
Michelson. 




Pi Pz Pz 

Fig. 13C/. Special Michelson interferometer used in accurately comparing the wave- 
length of light with the standard meter. 

etalons were first mounted in an interferometer of special design (Fig. 
13C7), with a field of view covering the four mirrors, M\, M 2 , M[, and M' 2 . 
With the aid of the white light fringes the distances of M, M x , and 
M\ from the eye were made equal, as shown in the figure. Substituting 
the light of one of the cadmium lines for white light, M was then moved 



INTERFERENCE OF TWO BEAMS OF LIGHT 255 

slowly from A to B, counting the number of fringes passing the cross 
hair. The count was continued until M reached the position B, which 
was exactly coplanar with M 2 , as judged by the appearance of the white- 
light fringes in the upper mirror of the shorter etalon. The fraction of 
a cadmium fringe in excess of an integral number required to reach 
this position was determined, giving the distance M\M% in terms of 
wavelengths. The shorter etalon was then moved through its own length, 
without counting fringes, until the white-light fringes reappeared in Mi. 
Finally M was moved to C, when the white-light fringes appeared in M' 2 
as well as in M 2 . The additional displacement necessary to make M 
coplanar with M 2 was measured in terms of cadmium fringes, thus 
giving the exact number of wavelengths in the longer etalon. This 
was in turn compared with the length of a third etalon of approximately 
twice the length of the second, by the same process. 

The length of the largest etalon was about 10 cm. This was finally 
compared with the prototype meter by alternately centering the white- 
light fringes in its upper and lower mirrors, each time the etalon was 
moved through its own length. Ten such steps brought a marker on the 
side of the etalon nearly into coincidence with the second fiducial mark 
on the meter, and the slight difference was evaluated by counting cad- 
mium fringes. The 10 steps involve an accumulated error which does 
not enter in the intercomparison of the etalons, but nevertheless this 
was smaller than the uncertainty in setting on the end marks. 

The final results were, for the three cadmium lines: 

Red line 1 m = 1,553, 163.5X or X = 6438.4722 A 

Green line 1 m = 1,966,249.7\ or X = 5085.8240 A 

Blue line 1 m = 2,083,372. IX or X - 4799.9107 A 

Not only has the standard meter been determined in terms of what 
we now believe to be an invariable unit, the wavelength of light, but we 
have also obtained absolute determinations of the wavelength of three 
spectrum lines, the red line of which is at present the primary standard 
in spectroscopy. More recent measurements on the red cadmium line 
have been made (see Sec. 14.11). It now is internationally agreed that 
in dry atmospheric air at 15°C and a pressure of 760 mm Hg the red 
cadmium line, produced under the conditions described by Michelson, 
has the wavelength 

X r = 6438.4696 A 

A still more satisfactory line for use as a standard of wavelength 
has now been discovered,* namely, the green line of mercury as emitted 
by the single isotope, Hg 198 . This kind of mercury can be produced 
entirely free of the other mercury isotopes by the bombardment of gold 

* J. H. Wiens and L. W. Alvarez, Phys. Rev., 58, 1005, 1940. 



256 PHYSICAL OPTICS 

with neutrons. The line is considerably sharper than the cadmium 
standard, and its wavelength has been measured* as 5400.7532 A. Prob- 
ably it will replace XG438 as the primary standard of wavelength. 

13.14. Twyman and Green Interferometer. If a Michelson interfer- 
ometer is illuminated with strictly parallel monochromatic light, produced 
by a point source at the principal focus of a well-corrected lens, it becomes 
a very powerful instrument for testing the perfection of optical parts such 
as prisms and lenses. The piece to be tested is placed in one of the light 
beams, and the mirror behind it is so chosen that the reflected waves, 
after traversing the test piece a second time, again become plane. These 
waves are then brought to interference with the plane waves from the 
other arm of the interferometer by another lens, at the focus of which 
the eye is placed. If the prism or lens is optically perfect, so that the 
returning waves are strictly plane, the field will appear uniformly illumi- 
nated. Any local variation of the optical path will, however, produce 
fringes in the corresponding part of the field, which are essentially the 
"contour lines" of the distorted wave front. Even though the surfaces 
of the test piece may be accurately made, the glass may contain regions 
that are slightly more or less dense. With the Twyman and Green 
interferometer these may be detected, and corrected for by local polishing 
of the surface.! 

13.15. Index of Refraction by Interference Methods. If a thickness t 
of a substance having an index of refraction n is introduced into the path 
of one of the interfering beams in the interferometer, the optical path 
in this beam is increased because of the fact that light travels more 
slowly in the substance and consequently has a shorter wavelength. 
The optical path (Eq. 11/) is now nt through the medium, whereas it was 
practically t through the corresponding thickness of air {n = 1). Thus 
the increase in optical path due to insertion of the substance is (n — 1)2. J 
This will introduce (n — 1)//X extra waves in the path of one beam; so if 
we call Am the number of fringes by which the fringe system is displaced 
when the substance is placed in the beam, we have 

(n - l)t - (Am) A (13fc) 

In principle a measurement of Am, t, and X thus gives a determination 
of n. 

In practice, the insertion of a plate of glass in one of the beams pro- 

* W. F. Meggers and F. O. Westfall, J. Research Natl. Bur. Standards, 44, 447-455, 
1950. 

t For a more complete description of the use of this instrument, see F. Twyman, 
"Prism and Lens Making," 2d ed., chap. 12, Hilger and Watts, London, 1952. 

X In the Michelson interferometer, where the beam traverses the substance twice 
in its back-and-forth path, t is twice the actual thickness. 



INTERFERENCE OF TWO BEAMS OF LIGHT 



257 



duces a discontinuous shift of the fringes so that the number Am cannot 
be counted. With monochromatic fringes it is impossible to tell which 
fringe in the displaced set corresponds to one in the original set. With 
white light, the displacement in the fringes of different colors is very 
different because of the variation of n with wavelength, and the fringes 
disappear entirely. This illustrates the necessity of the compensating 
plate Gi in Michelson's interferometer if white-light fringes are to be 
observed. If the plate of glass is very thin, these fringes may still be 
visible, and this affords a method of measuring n for very thin films. 
For thicker pieces, a practicable method is to use two plates of identical 



_ S 




Fig. 13F. (a) The Jam in and (6) the Mach-Zehnder interferometers. 



thickness, one in each beam, and to turn one gradually about a vertical 
axis, counting the number of monochromatic fringes for a given angle 
of rotation. This angle then corresponds to a certain known increase 
in effective thickness. 

For the measurement of the index of refraction of gases, which can be 
introduced gradually into the light path by allowing the gas to flow into 
an evacuated tube, the interference method is the most practicable one. 
Several forms of refractometers have been devised especially for this 
purpose, of which we shall describe three, the Jamin, the Mach-Zehnder, 
and the Rayleigh refractometers. 

Jamin's refractometer is shown schematically in Fig. 137(a). Mono- 
chromatic light from a broad source S is broken into two parallel beams 
(1) and (2) by reflection at the two parallel faces of a thick plate of 
glass Gi. These two rays pass through to another identical plate of 
glass G2 to recombine after reflection, forming interference fringes known 
as Brewster's fringes (see Sec. 14.11). If now the plates are parallel, the 
light paths will be identical. Suppose as an experiment we wish to 
measure the index of refraction of a certain gas at different temperatures 



258 PHYSICAL OPTICS 

and pressures. Two similar evacuated tubes 7\ and 7\ of equal length 
are placed in the two parallel beams. Gas is slowly admitted to tube 
Ti. Counting the number of fringes Am crossing the field while the gas 
reaches the desired pressure and temperature, the value of n can be found 
by applying Eq. 13/c. It is found experimentally that at a given tem- 
perature the value (n — 1) is directly proportional to the pressure. This 
is a special case of a theoretical law known as the Lorenz-Lorentz* law 
according to which 

TO = (» " i) &t8 - "■* x » ™ 

Here p is the density of the gas. When n is very nearly unity, the factor 
in + l)/(n 2 + 2) is nearly constant, as required by the above experi- 
mental observation. 

The interferometer devised by Mach and Zehnder, and shown in Fig. 
137(b), has a similar arrangement of light paths, but they may be much 
farther apart. The role of the two glass blocks in the Jamin instrument 
is here taken by two pairs of mirrors, the pair M i and Mi functioning 
like G\, and the pair M 3 and Mi like (t 2 . The first surface of Mi and the 
second surface of M 4 are half-silvered. Although it is more difficult to 
adjust, the Mach-Zehnder interferometer is the only one suitable for 
studying slight changes of refractive index over a considerable area and 
is used, for example, in measuring the flow patterns in wind tunnels. 
Contrary to the situation in the Michelson interferometer, the light 
traverses a region such as T in the figure in only one direction, a fact 
which simplifies the study of local changes of optical path in that region. 

The purpose of the compensating plates Ci and C 2 in Figs. 13F(a) and 
13W is to speed up the measurement of refractive index. As the two 
plates, of equal thickness, are rotated together by the single knob attached 
to the dial D, one light path is shortened and the other lengthened. The 
device can therefore compensate for the path difference in the two tubes. 
The dial, if previously calibrated by counting fringes, can be made to 
read the index of refraction directly. The sensitivity of this device 
can be varied at will, a high sensitivity being obtained when the angle 
between the two plates is small, and a low sensitivity when the angle 
is large. 

* H. A. Lorentz (1853-1928). For many years professor of mathematical physics 
at the University of Leyden, Holland. Awarded the Nobel prize (1902) for his work 
on the relations between light, magnetism, and matter, he also contributed notably to 
other fields of physics. Gifted with a charming personality and kindly disposition, 
he traveled a great deal, and was widely known and liked. By a strange coincidence 
L. Lorenz of Copenhagen derived the above law from the elastic-solid theory only a 
few months before Lorentz obtained it from the electromagnetic theory. 



INTERFERENCE OF TWO BEAMS OF LIGHT 



259 



In Rayleigh's* refractometer (Fig. 13 W) monochromatic light from a 
linear source 8 is made parallel by a lens L\ and split into two beams by 
a fairly wide double slit. After passing through two exactly similar 
tubes and the compensating plates, these are brought to interfere by the 



F, 





Fig. 13Tf. Rayleigh's refractometer. 

lens L 2 . This form of refractometer is often used to measure slight 
differences in refractive index of liquids and solutions. 



PROBLEMS 

1. Young's experiment is performed with light of the green mercury line. If the 
fringes are measured with a micrometer eyepiece 80 cm behind the double slit, it is 
found that 20 of them occupy a distance of 10.92 mm. Find the distance between 
the two slits. 

2. A double slit of separation \ mm is illuminated by light of the blue cadmium 
line. How far behind the slits must one go to obtain fringes that are 1 mm apart? 

Ans. 104.2 cm. 

3. Describe what would be observed if a double slit were illuminated by light of the 
two yellow mercury lines, XX5769 and 5790 A. Assuming the two lines to be perfectly 
sharp and of equal intensity, calculate the visibility of the fringes near m = 50. 

4. In Young's double-slit experiment, when a thin film of transparent material is 
placed over one of the slits, the central bright fringe of the white-light fringe system 
is displaced by 3.6 fringes. The refractive index of the material is 1.40, and the 
effective wavelength of the light 5500 A. (a) By how much does the film increase 
the optical path? (b) What is the exact thickness of the film? (c) What would 
probably be observed if a piece of the material 1 mm thick were used, and why? 

Ans. (a) 1.98 X 10" 4 cm. (6) 4.95 X 10"* cm. (c) No fringes. 

5. Lloyd's mirror is easily demonstrated with microwaves, using as a reflector a 
sheet of metal lying on a table. If the source has a frequency of 10,000 Mc and is 
placed 6 cm above the surface of the table, find the height above this surface of the 
first two maxima, 4 m away from the source. 

6. In Fresnel's biprism and mirrors, coherent parts of the two virtual sources are in 
corresponding positions, whereas in Lloyd's mirror they are inverted with respect to 
each other. What effect will this difference have on the appearance of the fringes 
produced when the source slit is not extremely narrow? 

.4ns. Fringes of higher order become indistinct in Lloyd's mirror. 

* Lord Rayleigh (third Baron) (1842-1919). Professor of physics at Cambridge 
University and the Royal Institution of Great Britain. Gifted with great mathe- 
matical ability and physical insight, he made important contributions to many fields 
of physics. His works on sound and on the scattering of light (Sec. 22.9) are the best 
known. He was a Nobel prize winner in 1904. 



2G0 PHYSICAL OPTICS 

7. A Fresnel bipriam is to be constructed for use on a 2-m optical bench. The 
source slit is to be at one end of the bench, and the eyepiece at the other. Because 
of the finite width of the source slit, it is not permitted to place the biprism less than 
50 cm from it. Find the value of the refracting angles of the biprism necessary to 
produce sodium fringes having a separation of 0.8 mm, if the glass has n = 1.55. 

8. A Fresnel biprism with apex angles 1°30' is used to form interference fringes. 
The refractive index is 1.52. Find the fringe separation for red light, X6503, when the 
distance between the source and the prism is 20 cm and that between the prism and 
the screen 80 cm. Ans. 0.1205 mm. 

9. With Fresnel's mirrors, what must be the angle between the mirrors, in degrees, 
in order to produce sodium fringes 0.5 mm apart, if the slit is 30 cm from the inter- 
section of the mirrors and the screen 120 cm from the slit? 

10. It is desired to determine the unknown concentration of a solution to ± 0.002 per 
cent by comparing its refractive index with that of a standard solution in the Rayleigh 
refractometer using a sodium lamp as a source. A 5 per cent standard solution has 
n = 1.4316, a 10 per cent solution n = 1.4425, and n varies linearly with concentra- 
tion between these values. What lengths should the tubes have in order to achieve 
the required accuracy, assuming that one can estimate fringe displacements to one- 
twentieth of a fringe? Ans. 6.76 mm. 

11. How far must the movable mirror of a Michelson interferometer be displaced 
for 2000 fringes of the green cadmium line to cross the center of the field of view? 

12. Find the angular radius of the sixth bright fringe in a Michelson interferometer 
when the central path difference (2d) is 4 mm, and when it is 30 mm. Assume blue 
light of wavelength 4358 A and that the interferometer is adjusted in each case so 
that the first bright fringe forms a maximum at the center of the pattern. 

Ans. 1.892°. 0.690°. 

13. Investigate the effect on the white-light fringes of placing a thin sheet of tele- 
scope crown glass in one arm of a Michelson interferometer. The refractive indices 
of this glass are given in Table 23-1. Assuming that the colored fringes disappear 
at a path difference equal to six wavelengths of sodium light, what would be the 
maximum allowable thickness of the glass sheet in order to see any fringes when it is 
inserted? 

14. Prove that the increase in optical path produced by rotating a plane-parallel 
compensating plate of thickness t and index n through an angle <t> from the perpen- 
dicular is given by 

A = l(\/n 2 — sin 2 — cos $ — n + 1) 

(Hint: Take account of the change of path through air, as well as through glass.) 

15. The two compensating plates of a Jamin refractometer are fixed at an angle 
of 5° with each other. One plate is vertical when the fringes are first observed. 
Through what angle should the pair be rotated to produce a shift of 20 fringes of green 
light, X = 5500 A, the refractive index being 1.500? Assume that the plates are 
turned toward the symmetrical position. 

16. For a spectrum line having the contour due to Doppler broadening, it can be 
proved that the path difference at which the visibility curve falls to its half value is 
2.77 /Ak, where Ak is the width of the line at half maximum, expressed in propagation 
numbers. From the data given in Sec. 13.12, calculate the width in angstroms of 
the red cadmium line. Ans. 0.018 A. 

17. The two tubes of a Jamin interferometer are each 40 cm long. One is evacu- 
ated, and the other contains argon at atmospheric pressure. The refractive index 
of the latter is 1.00028. How many fringes of the green mercury line would be counted 
when the argon is pumped out? 






CHAPTER 14 

INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 



Some of the most beautiful effects of interference result from the mul- 
tiple reflection of light between the two surfaces of a thin film of trans- 
parent material. These effects require no special apparatus for their 
production or observation and are familiar to anyone who has noticed 
the colors shown by thin films of oil on water, by soap bubbles, or by 
cracks in a piece of glass. We begin our investigation of this class of 
interference by considering the somewhat idealized case of reflection from 
a film with perfectly plane sides which are parallel to each other. 




Fig. 14A. Multiple reflections in a plane-parallel film. 

14.1. Reflection from a Plane -parallel Film. Let a ray of light from 
a source S be incident on the surface of such a film at A (Fig. 14 A). 
Part of this will be reflected as ray (1) and part refracted in the direction 
AF. Upon arrival at F, part of the latter will be reflected to B and part, 
refracted toward H. At B the ray FB will be again divided. A continu- 
ation of this process yields two sets of parallel rays, one on each side 
of the film. In each of these sets, of course, the intensity decreases 
rapidly from one ray to the next. If the set of parallel reflected rays 
is now collected by a lens and focused at the point P, each ray will have 
traveled a different distance, and the phase relations may be such as to 

261 



262 



PHYSICAL OPTICS 



produce destructive or constructive interference at that point. It is 
such interference that produces the colors of thin films when they are 
viewed by the naked eye. In such a case L is the lens of the eye, and P 
lies on the retina. 

In order to find the phase difference between these rays, we must first 
evaluate the difference in the optical path traversed by a pair of succes- 
sive rays, such as rays (1) and (2). In Fig. 14J5 let d be the thickness of 

the film, n its index of refraction, 
X the wavelength of the light, and 
<j> and <j>' the angles of incidence 
and refraction. If BD is perpen- 
dicular to ray (1), the optical paths 
from D and B to the focus of the 
lens will be equal. Starting at A, 
ray (2) has the path AFB in the 
film and ray (1) the path AD in 
air. The difference in these optical 
paths (Eq. 11/) is given by 




A = n(AFB) -AD 



Fig. 145. Optical path difference be- 
tween two consecutive rays in multiple 
reflection (see Fig. 144). 



If BF is extended to intersect the per- 
pendicular line AE at G, AF = GF 
because of the equality of the angles 
of incidence and reflection at the lower surface. Thus we have 



A = n(GB) - AD = n(GC + CB) - AD 

Now AC is drawn perpendicular to FB; so the broken lines AC and 
DB represent two successive positions of a wave front reflected from 
the lower surface. The optical paths, as was shown in Sec. 11.6, must 
be the same by any ray drawn between two wave fronts; so we may write 

n(CB) = AD 



The path difference then reduces to 

A = n(GC) = n(2d cos <*>') 



(14a) 



If this path difference is a whole number of wavelengths, we might 
expect rays (1) and (2) to arrive at the focus of the lens in phase with 
each other and produce a maximum of intensity. However, we must 
take account of the fact that ray (1) undergoes a phase change of tt at 
reflection, while ray (2) does not, since it is internally reflected (Sec. 11.8). 
The condition 

2nd cos 4>' = raX mini»&a (146) 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 263 

then becomes a condition for destructive interference as far as rays (1) 
and (2) are concerned. As before, m = 0, 1, 2, ... is the order of 
interference. 

Next we examine the phases of the remaining rays, (3), (4), (5), . . . . 
Since the geometry is the same, the path difference between rays (3) 
and (2) will also be given by Eq. 14a, but here there are only internal 
reflections involved, so that if Eq. 146 is fulfilled, ray (3) will be in the 
same phase as ray (2). The same holds for all succeeding pairs, and so 
we conclude that under these conditions rays (1) and (2) will be out of 




Fig. 14C. Amplitudes of successive rays in multiple reflection. 



phase, but rays (2), (3), (4), . . . , will be in phase with each other. 
On the other hand, if conditions are such that 

2nd cos 0' = (m + £)X maxima (14c) 

ray (2) will be in phase with (1), but (3), (5), (7), ... will be out of 
phase with (2), (4), (6), . . . . Since (2) is more intense than (3), (4) 
more intense than (5), etc., these pairs cannot cancel each other, and 
since the stronger series combines with (1), the strongest of all, there 
will be a maximum of intensity. 

For the minima of intensity, ray (2) is out of phase with ray (1), but 
(1) has a considerably greater amplitude than (2), so that these two will 
not completely annul each other. We can now prove that the addition 
°f (3), (4), (5), . . . , which are all in phase with (2), will give a net 
amplitude just sufficient to make up the difference and to produce com- 
plete darkness at the minima. Using a for the amplitude of the incident 
wave, r for the fraction of this reflected, and t or t' for the fraction trans- 
mitted in going from rare to dense or dense to rare, as was done in Stokes' 
treatment of reflection in Sec. 11.8, Fig. 14C is constructed and the ampli- 
tudes labeled as shown. In accordance with Eq. lip, we have taken the 
fraction reflected internally and externally to be the same. Adding 
the amplitudes of all the reflected rays but the first on the upper side 



264 PHYSICAL OPTICS 

of the film, we obtain the resultant amplitude, 

A = atrt' ■+ atrH' + atrH' 4- atrH' + • • • 
= atrt' (I + r 2 + r* + r 6 + • • •) 

Since r is necessarily less than 1, the geometrical series in parentheses 
has a finite sum equal to 1/(1 — r 2 ), giving 

A = atrt' 



(1 - r*) 

But from Stokes' treatment (Eq. llo), W = 1 — r 2 ; so we obtain finally 

A = ar (14d) 

This is just equal to the amplitude of the first reflected ray, so we con- 
clude that under the conditions of Eq. 146 there will be complete destruc- 
tive interference. 

14.2. Fringes of Equal Inclination. If the image of an extended source 
reflected in a thin plane-parallel film be examined, it will be found to be 
crossed by a system of distinct interference fringes, provided the source 
emits monochromatic light and provided the film is sufficiently thin. 
Each bright fringe corresponds to a particular path difference giving an 
integral value of m in Eq. 14c. For any fringe, the value of <f> is fixed; 
so the fringe will have the form of the arc of a circle whose center is at 
the foot of the perpendicular drawn from the eye to the plane of the film. 
Evidently we are here concerned with fringes of equal inclination, and 
the equation for the path difference has the same form as for the circular 
fringes in the Michelson interferometer (Sec. 13.9). 

The necessity of using an extended source will become clear upon con- 
sideration of Fig. 14 A. If a very distant point source S is used, the 
parallel rays will necessarily reach the eye at only one angle (that required 
by the law of reflection), and will be focused to a point P. Thus only 
one point will be seen, either bright or dark, according to the phase 
difference at this particular angle. It is true that, if the source is not 
very far away, its image on the retina will be slightly blurred, because 
the eye must be focused for parallel rays to observe the interference. 
The area illuminated is small, however, and in order to see an extended 
system of fringes, we must obviously have many points S, spread out in a 
broad source so that the fight reaches the eye from various directions. 

These fringes are seen by the eye only if the film is very thin, unless 
the light is reflected practically normal to the film. At other angles, 
since the pupil of the eye has a small aperture, increasing the thickness 
of the film will cause the reflected rays to get so far apart that only one 
enters the eye at a time. Obviously no interference can occur under 
these conditions. Using a telescope of large aperture, the lens may 
include enough rays for the fringes to be visible with thick plates, but 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 



265 



unless viewed nearly normal to the plate, they will be so finely spaced 
as to be invisible. The fringes seen with thick plates near normal 
incidence are often called Haidinger* fringes. 

14.3. Interference in the Transmitted Light. The rays emerging 
from the lower side of the film, shown in Fig. 14A and 14C, may also 
be brought together with a lens and caused to interfere. Here, however, 
there are no phase changes at reflection for any of the rays, and the 
relations are such that Eq. 146 now becomes the condition for maxima 
and Eq. 14c the condition for minima. For maxima the rays u, v, 
w, . . . of Fig. 14 A are all in phase, 
while for minima v, x, . . . are out 
of phase with u, w, . . . . When 
the reflectance r 2 has a low value, 
as is the case with the surfaces of 
unsilvered glass, the amplitude of u 
is much the greatest in the series, 
and the minima are not by any 
means black. Figure 14D shows 
quantitative curves for the inten- 
sity transmitted, It, and reflected, 
In, plotted in this instance for 
r = 0.2 according to Eqs. 14j and 
14fc, ahead. The corresponding re- 
flectance of 4 per cent is closely 
that of glass at normal incidence. 
The abscissas 5 in the figure repre- 
sent the phase difference between successive rays in the transmitted set, 
or between all but the first pair in the reflected set, which by Eq. 14a is 




2x 4* 6*' 

Phase angle & 

Fig. 14D. Intensity contours of the 
reflected and transmitted fringes from a 
film having a reflectance of 4 per cent. 



8 = kA = — A = — nd cos <f>' 

A A 



(14e) 



It will be noted that the curve for I R looks very much like the cos 2 contour 
obtained from the interference of two beams. It is not exactly the same, 
however, and the resemblance holds only when the reflectance is small. 
Then the first two reflected beams are so much stronger than the rest 
that the latter have little effect. The important changes that come in 
at higher values of the reflectance will be discussed in Sec. 14.7. 

14.4. Fringes of Equal Thickness. If the film is not plane parallel, so 
that the surfaces make an appreciable angle with each other as in Fig. 
142? (a), the interfering rays do not enter the eye parallel to each other, 

* W. K. Haidinger (1795-1871). Austrian mineralogist and physicist, for 17 years 
director of the Imperial Geological Institute in Vienna. 



266 



PHYSICAL OPTICS 



but appear to diverge from a point near the film. The resulting fringes 
resemble the localized fringes in the Michelson interferometer, and appear 
to be formed in the film itself. If the two surfaces are plane, so that the 
film is wedge-shaped, the fringes will be practically straight following 
the lines of equal thickness. In this case the path difference for a given 




Spacer- 




ib) 



Fig. 14E. Fringes of equal thickness, (a) Method of visual observation. (6) 
Photograph taken with a camera focused on the plates. 

pair of rays is practically that given by Eq. 14a. Provided that obser- 
vations are made almost normal to the film, the factor cos <f>' may be 
considered equal to 1, and the condition for bright fringes becomes 



2nd = (m + £)X 



(14/) 



In going from one fringe to the next m increases by 1, and this requires 
that the optical thickness of the film, nd, should change by A/2. 

Fringes formed in thin films are easily shown in the laboratory or lecture 
room by using two pieces of ordinary plate glass. If they are laid together 
with a thin strip of paper along one edge, we obtain a wedge-shaped film 
of air between the plates. When a sodium flame or arc is viewed as in Fig. 
HE, yellow fringes are clearly seen. If a carbon arc and filter are used, 
the fringes may be projected on a screen with a lens. On viewing the 
reflected image of a monochromatic source, one will find it to be crossed 
by more or less straight fringes, such as those in Fig. 14/2(6). 

This class of fringes has important practical applications in the testing 
of optical surfaces for planeness. If an air film is formed between two 
surfaces, one of which is perfectly plane and the other not, the fringes 
will be irregular in shape. Any fringe is characterized by a particular 
value of m in Eq. 14/, and hence will follow those parts of the film where 
d is constant. That is, the fringes form the equivalent of contour lines 
for the uneven surface. The contour interval is X/2, since for air n = 1, 
and going from one fringe to the next corresponds to increasing d by this 
amount. The standard method of producing optically plane surfaces 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 



267 



uses repeated observation of the fringes formed between the working 
surface and an optical flat, the polishing being continued until the fringes 
are straight. In Fig. 14E(b) it will be noticed that there is considerable 
distortion of one of the plates near the bottom. 

14.5. Newton's Rings. If the fringes of equal thickness are produced 
in the air film between a convex surface of a long-focus lens and a plane 
glass surface, the contour lines will be circular. The ring-shaped fringes 
thus produced were studied in detail by Newton,* although he was not 
able to explain them correctly. 
For purposes of measurement, the 
observations are usually made at 
normal incidence by an arrange- 
ment such as that in Fig. 14F, 
where the glass plate G reflects the 
light down on the plates. After 
reflection, it is transmitted by G 
and observed in the low-power 
microscope T. Under these con- 
ditions the positions of the max- 
ima are given by Eq. 14/, where 
d is the thickness of the air film. 
Now if we designate by R the 
radius of curvature of the surface A, and assume that A and B are just 
touching at the center, the value of d for any ring of radius r is the sagitta 
of the arc, given by 




Fig. \AF. Experimental arrangement used 
in viewing and measuring Newton's rings. 



J r 



(140) 



Substitution of this value in Eq. 14/ will then give a relation between the 
radii of the rings and the wavelength of the light. For quantitative 
work, one may not assume the plates to barely touch at the point of 
contact, since there will always be either some dust particles or distortion 
by pressure. Such disturbances will merely add a small constant to 
Eq. 14g, however, and their effect may be eliminated by measuring the 
diameters of at least two rings. 

Because the ring diameters depend on wavelength, white light will 

* Isaac Newton (1G42-1727). Besides laying foundations of the science of mechan- 
ics, Newton devoted considerable time to the study of light and embodied the results 
in his famous "Opticks." It seems strange that one of the most striking demon- 
strations of the interference of light, Newton's rings, should be credited to the chief 
proponent of the corpuscular theory of light. N iwton'a advocacy of the corpuscular 
theory was not so uncompromising as it is generally represented. This is evident 
to anyone consulting his original writings. The original discovery of Newton's 
rings is now attributed to Robert Hooke. 



268 



PHYSICAL OPTICS 



produce only a few colored rings near the point of contact. With mono- 
chromatic light, however, an extensive fringe system such as that shown 
in Fig. 14G(a) is observed. When the contact is perfect, the central 
spot is found to be black. This is direct evidence of the relative phase 
change of -n between the two types of reflection, air-to-glass and glass-to- 
air, mentioned in Sec. 14.1. If there were no such phase change, the rays 
reflected from the two surfaces in contact should be in the same phase, 




(a) (b) 

FlG. 14G. Newton's rings (a) by reflection; (6) by transmission. 

and produce a bright spot at the center. In an interesting modification 
of the experiment, due to Thomas Young, the lower plate has a higher 
index of refraction than the lens, and the film between is filled with 
an oil of intermediate index. Then both reflections are at "rare-to- 
dense" surfaces, no relative phase change occurs, and the central fringe 
of the reflected system is bright. The experiment does not tell us at 
which surface the phase change in the ordinary arrangement occurs, but 
it is now definitely known (Sec. 25.4) that it occurs at the lower (air-to- 
glass) surface. 

A ring system is also observed in the light transmitted by the Newton 's- 
ring plates. These rings are exactly complementary to the reflected ring 
system, so that the center spot is now bright. The contrast between 
bright and dark rings is small, for reasons already discusse 1 in Sec. 14.3. 
A reproduction of the transmitted pattern is shown in Fig 14G(b). 

14.6. Nonreflecting Films. A simple and very importa it application 
of the principles of interference in thin films has been the production of 
the so-called coated surfaces. If a film of a transpar nt substance 
of refractive index n' be deposited on glass of a larger index n, to a thick- 
ness of one-quarter of the wavelength of light in the film, so that 

w X 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 2b9 

the light reflected at normal incidence is almost completely suppressed 
by interference. This corresponds to the condition m = in Eq. 14c, 
which here becomes a condition for minima because the reflections at 
both surfaces are "rare-to-dense." The waves reflected from the lower 
surface have an extra path of one-half wavelength over those from the 
upper surface, and the two, combined with the weaker waves from mul- 
tiple reflections, therefore interfere destructively. For the destruction 
to be complete, however, it is necessary that the fraction of the ampli- 
tude reflected at each of the two surfaces be exactly the same, since this 
specification is made in proving the relation of Eq. 14d. It will be true 
for a film in contact with a medium of higher index only if the index of 
the film obeys the relation 

n' = \/n 

This can be proved from Eq. 25e of the chapter on reflection by substi- 
tuting n' for the refractive index of the upper surface and n/n' for that 
of the lower. Similar considerations will show that such a film will give 
zero reflection from the glass side as well as from the air side. Of course 
no light is destroyed by a nonreflecting film; there is merely a redistribu- 
tion such that a decrease of reflection carries with it a corresponding 
increase of transmission. 

The practical importance of these films is that by their use one can 
greatly reduce the loss of light by reflection at the various surfaces of a 
system of lenses or prisms. Stray light reaching the image as a result 
of these reflections is also largely eliminated, with a resulting increase 
in contrast. Almost all optical parts of high quality are now coated 
to reduce reflection. The coatings were first made by depositing several 
monomolecular layers of an organic substance on glass plates. More 
durable ones are now made by evaporating calcium or magnesium fluo- 
ride on the surface in vacuum, or by chemical treatment with acids which 
leave a thin layer of silica on the surface of the glass. Properly coated 
lenses have a purplish hue by reflected light. This is a consequence of 
the fact that the condition for destructive interference can be fulfilled 
for only one wavelength, which is usually chosen to be one near the 
middle of the visible spectrum. The reflection of red and violet light 
is then somewhat larger. Furthermore, coating materials of sufficient 
durability have too high a refractive index to fulfill the condition stated 
above. Considerable improvement in these respects can be achieved by 
using two or more superimposed layers, and such films are capable of 
reducing the total reflected light to one-tenth of its value for the uncoated 
glass. This refers, of course, to light incident perpendicularly on the 
surface. At other angles, the path difference will change because of the 
factor cos <b' in Eq. 14a. Since, however, the cosine does not change 
rapidly in the neighborhood of 0°, the reflection remains low over a fairly 



270 



PHYSICAL OPTICS 



large range of angles about the normal. The multiple films, now called 
multilayers, may also be used, with suitable thicknesses, to accomplish the 
opposite purpose, namely, to increase the reflectance. They may be 
used, for example, as "beam-splitting" mirrors to divide a beam of light 
into two parts of a given intensity ratio. The division can thus be 
accomplished without the losses of energy by absorption that are inherent 
in the transmission through, and reflection from, a thin metallic film. 

14.7. Sharpness of the Fringes. As the reflectance of the surfaces is 
increased, either by the above method or by lightly silvering them, the 
fringes due to multiple reflections become much narrower. The striking 




100% 



Fig. HH. Intensity contours of fringes due to multiple reflections, showing how the 
sharpness depends on reflectance. 

changes that occur are shown in Fig. 14//, which is plotted for r 2 = 0.04, 
0.40, and 0.80 according to the theoretical equations to be derived below. 
The curve labeled 4% is just that for unsilvered glass which was given 
in Fig. 14Z). Since, in the absence of any absorption, the intensity 
transmitted must be just the complement of that reflected, the same 
plot will represent the contour of either set. One is obtained from the 
other by merely turning the figure upside down, or by inverting the scale 
of ordinates, as is shown at the right in Fig. 14//. 

In order to understand the reason for the narrowness of the transmitted 
fringes when the reflectance is high, we may use the graphical method of 
compounding amplitudes already discussed in Sees. 12.2 and 13.4. Refer- 
ring back to Fig. 14C we notice that the amplitudes of the transmitted rays 
are given by alt' , att'r 2 , att'r*, . . . , or in general for the mth ray by 
att'r 2m . We thus have to find the resultant of an infinite number of 
amplitudes which decrease in magnitude more rapidly the smaller the 
fraction r. In Fig. l4/(a) the magnitudes of the amplitudes of the first 
10 transmitted rays are drawn to scale for the 50 per cent and 80 per cent 
cases in Fig. 14//, i.e., essentially for r = 0.7 and 0.9. Starting at any 
principal maximum, with S = 2irm, these individual amplitudes will all 
be in phase with each other, so the vectors are all drawn parallel to give a 
resultant that has been made equal for the two cases. If we now go 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 271 

slightly to one side of the maximum, where the phase difference introduced 
between successive rays is ir/10, each of the individual vectors must be 
drawn making an angle of x/10 with the preceding one, and the resultant 
found by joining the tail of the first to the head of the last. The result is 
shown in diagram (6). It will be seen that in the case r = 0.9, in which 
the individual amplitudes are much more nearly equal to each other, 
the resultant R is already considerably less than in the other case. In 
diagram (c), where the phase has changed by jt/5, this effect is much 
more pronounced; the resultant has fallen to a considerably smaller value 

T = 0.7 r = 0.9 
6 = 2itm ■ • 1 — •— - *• (a) » . ■ ■ — » 






16) 



(c) 

t = 2vm+f 

Fig. 14/. Graphical composition of amplitudes for the first 10 multiply reflected rays, 
with two difference reflectances. 

in the right-hand picture. Although a correct picture would include an 
infinite number of vectors, the later ones will have vanishing amplitudes, 
and we would reach a result similar to that found with the first 10. 

These qualitative considerations may be made more precise by deriving 
an exact equation for the intensity. To accomplish this, we must find 
an expression for the resultant amplitude A, the square of which deter- 
mines the intensity. Now A is the vector sum of an infinite series of 
diminishing amplitudes having a certain phase difference 8 given by Eq. 
14e. Here we may apply the standard method of adding vectors by first 
finding the sum of the horizontal components, then that of the vertical 
components, squaring each sum, and adding to get A 2 . In doing this, 
however, the use of trigonometric functions as in Sec. 12.1 becomes too 
cumbersome. Hence an alternative way of compounding vibrations, 
which is mathematically simpler for complicated cases, will be used. 

14.8. Method of Complex Amplitudes. In place of using the sine or 
the cosine to represent a simple harmonic wave, one may write the equa- 
tion in the exponential form* 

y = ae iC - u '- kx) *= ae^'e-* 

* For the mathematical background of this method, see E. T. Whittaker and G. N. 
Watson, "Modern Analysis," chap. 1, Cambridge University Press, New York, 1935. 



272 



PHYSICAL OPTICS 



where 8 = kx, and is constant at a particular point in space. The pres- 
ence of i = \/ — 1 in this equation makes the quantities complex. We 
may nevertheless use this representation, and at the end of the problem 
take either the real (cosine) or the imaginary (sine) part of the resulting 

expression. Now the time-varying 



V (Imaginary axis) 




(Real axis). 

Fig. 14/. Representation of a vector in 
the complex plane. 



factor exp (iut) is of no importance 
in combining waves of the same 
frequency, since the amplitudes and 
relative phases are independent of 
time. The other factor, a exp 
( — id), is called the complex ampli- 
tude. It is a complex number, 
whose modulus a is the real ampli- 
tude, and whose argument 5 gives 
the phase relative to some standard 



phase. The negative sign merely indicates that the phase is behind the 
standard phase. In general, the vector a is given by 

a = ae a = x + iy = a(cos 5 + i sin 5) 

Then it will be seen that 



o = \/x 2 -f- y 1 



tan 5 = - 
x 



Thus if a is represented as in Fig. 14J, plotting horizontally its real part 
and vertically its imaginary part, it will have the magnitude a and will 
make the angle 5 with the x axis, as we require for vector addition. 

The advantage of using complex amplitudes lies in the fact that the 
algebraic addition of two or more is equivalent to vector addition of the 
real amplitudes. Thus for two such quantities 



so that if 
and 



Ae ie = aie ai + a 2 e a ' 

#i + x-2 = ai cos 6i + a2 cos 82 = X 
V\ 4- y% = a\ sin h x + 02 sin 5 2 = Y 



it will be found that our previous Eqs. I2d and 12e require that 
A = \/X~ 2 + : ~Y* 



n Y 
tan 6 = y 



(14/i) 



Thus, to get a vector sum, we need only obtain the algebraic sums X = Sx,- 
and Y = 2?/, of the real and imaginary parts, respectively, of the complex 
amplitudes. In obtaining the resultant intensity as proportional to the 
square of the real amplitude, we multiply the resultant complex amplitude 
by its complex conjugate, which is the same expression with i replaced 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 273 

by —i throughout. The justification for this procedure follows from the 
relations 

(x + iY)(x - iY) - x* + y» = a* \ (Ull 

Ae i6 . Ae -ie = A 2 j v. ') 

14.9. Derivation of the Intensity Function. For the fringe system 
formed by the transmitted light, the sum of the complex amplitudes is 
(see Fig. 14C) 

Ae i8 = ait' + aU'r 2 e a + alt'r A e i2S 4- • • • 

= a(l - r 2 )(l + rV 8 + r 4 e iM + • • •) 

where (1 — r 2 ) has been substituted for tt', according to Stokes' relation 
(Eq. llo). The infinite geometric series in the second parentheses has 
the common ratio r 2 exp (i8), and has a finite sum because r* < 1. 
Summing the series, one obtains 

Ae <e = ad ~ r 2 ) 

By Eq. 14t, the intensity is the product of this quantity by its complex 
conjugate, which yields 

/ ^ a(l ~ r 2 ) a(l - r 2 ) = q 2 (l - r 2 ) 2 

T ~ 1 - r 2 e a 1 - r 2 e~ iS 1 - r 2 (e'* + e-*») 4- r 4 

Since (e* 4- e _,i )/2 = cos 5, and a 2 ~ 7 , the intensity of the incident 
beam, we obtain the result, in terms of real quantities only, as 

It ~ /o 1 - 2r 2 cos 8 4- r* = ~ Af 2 ~ I (14j) 

The main features of the intensity contours in Fig. 14// can be read 
from this equation. Thus at the maxima, where 5 = 2irm, we have 
sin 2 (5/2) = 0, and It = /o- When the reflectance r 2 is large, approach- 
ing unity, the quantity 4r 2 /(l — r 2 ) 2 will also be large, and even a small 
departure of 8 from its value for the maximum will result in a rapid drop 
of the intensity. 

For the reflected fringes it is not necessary to carry through the summa- 
tion, since we know from the conservation of energy that, if no energy 
is lost through absorption, 

Ib + It - 1 (14fc) 

The reflected fringes are complementary to the transmitted ones, and 
for high reflectances become narrow dark fringes. These can be used to 



274 



PHYSICAL OPTICS 



make more precise the study of the contour of surfaces.* If there is 
appreciable absorption on transmission through the surfaces, as will be 
the case if they are lightly silvered, one may no longer assume that 
Stokes' relations, nor Eq. 14fc, hold. Going back to the derivation of 
Eq. 14j, it will be found that in this case the expression for It must be 
multiplied by (tt') 2 /(l — r 2 ) 2 . Here it' and r 2 are essentially the frac- 
tions of the intensity transmitted and reflected, respectively, by a single 
surface. Where the surfaces are metallized, there will be slight differences 
between I and t', as well as small phase changes upon reflection. The 




Fig. 14K. Fabry-Perot interferometer E,E 2 set up to show the formation of circular 
interference fringes from multiple reflections. 



transmitted fringes may still be represented by Eq. 14 j, however, with 
an over-all reduction of intensity, and a correction to 8 which merely 
changes slightly the effective thickness of the plate. 

14.10. Fabry-Perot Interferometer. This instrument utilizes the 
fringes produced in the transmitted light after multiple reflection in the 
air film between two plane plates thinly silvered on the inner surfaces 
(Fig. 14K). Since the separation d between the reflecting surfaces is 
usually fairly large (from 0.1 to 10 cm) and observations are made near 
the normal direction, the fringes come under the class of fringes of equal 
inclination (Sec. 14.2). To observe the fringes, the light from a broad 
source (S1&2) of monochromatic light is allowed to traverse the inter- 
ferometer plates EiE 2 . Since any ray incident on the first silvered sur- 
face is broken by reflection into a series of parallel transmitted rays, it is 
essential to use a lens L, which may be the lens of the eye, to bring these 
parallel rays together for interference. In Fig. 14/C a ray from the point 
Pi on the source is incident at the angle 6, producing a series of parallel 
rays at the same angle, which are brought together at the point Po on 
the screen AB. It is to be noted that Pi is not an image of Pi. The 

*S. Tolansky, "Multiple-beam Interferometry," Oxford University Press, New 
Vork, 1948. 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 275 

condition for reinforcement of the transmitted rays is given by Eq. 146 
with n — 1 for air, and <£' = 0, so that 

2d cos 6 = m\ maxima (14J) 

This condition will be fulfilled by all points on a circle through Pi with 
its center at 0, the intersection of the axis of the lens with the screen 
AB. When the angle is decreased, the cosine will increase until another 
maximum is reached for which m is greater by 1, 2, . . . , so that we 
have for the maxima a series of concentric rings on the screen with 
as their center. Since Eq. 14/ is the same as Eq. 13<? for the Michelson 
interferometer, the spacing of the rings is the same as for the circular 
fringes in that instrument, and they will change in the same way with 
change in the distance d. In the actual interferometer one plate is fixed, 
while the other may be moved toward or away from it on a carriage 
riding on accurately machined ways by a slow-motion screw. 

14.11. Brewster's* Fringes. In a single Fabry-Perot interferometer it 
is not practicable to observe white-light fringes, since the condition of 
zero path difference occurs only when the two silvered surfaces are 
brought into direct contact. By the use of two interferometers in series, 
however, it is possible to obtain interference in white light, and the 
resulting fringes have had important applications. The two plane- 
parallel "air plates" are adjusted to exactly the same thickness, or else 
one to some exact multiple of the other, and the two interferometers are 
inclined to each other at an angle of 1° or 2°. A ray that bisects the 
angle between the normals to the two sets of plates can then be split 
into two, each of which after two or more reflections emerges, having 
traversed the same path. In Fig. 14L these two paths are drawn as 
separate for the sake of clarity, though actually the two interfering 
beams are derived from the same incident ray, and are superimposed 
when they leave the system. The reader is referred to Fig. 13 V, where 
the formation of Brewster's fringes by two thick glass plates in Jamin's 
interferometer is illustrated. A ray incident at any other angle than 
that mentioned above will give a path difference between the two emerg- 
ing ones which increases with the angle, so that a system of straight 
fringes is produced. 

The usefulness of Brewster's fringes lies chiefly in the fact that when 
they appear, the ratio of the two interferometer spacings is very exactly 

* Sir David Brewster (1781-1868). Professor of physics at St. Andrew's, and later 
principal of the University of Edinburgh. Educated for the church, he became 
interested in light through repeating Newton's experiments on diffraction. He made 
important discoveries in double refraction and in spectrum analysis. Oddly enough, 
he opposed the wave theory of light in spite of the great advances in this theory that 
were made during his lifetime. 



276 PHYSICAL OPTICS 

a whole number. Thus, in the redetermination of the length of the 
standard meter in terms of the wavelength of the red cadmium line, a 
series of interferometers was made, each having twice the length of the 
preceding, and these were intercompared using Brewster's fringes. The 
number of wavelengths in the longest, which was approximately 1 m 
long, could be found in a few hours by this method. It should finally 
be emphasized that this type of fringe results from the interference of 



(ft) 





Fig. 14L. Light paths for the formation of Brewster's fringes, (a) With two plates 
of equal thickness. (6) With one plate twice as thick as the other. The inclination 
of the two plates is exaggerated. 

only two beams, and therefore cannot be made very narrow, as can the 
usual fringes due to multiple reflections. 

14.12. Chromatic Resolving Power. The great advantage of the 
Fabry-Perot interferometer over the Michelson instrument lies in the 
sharpness of the fringes. Thus it is able to reveal directly those details 
of fine structure and line width that previously could only be inferred 
from the behavior of the visibility curves. The difference in the appear- 
ance of the fringes for the two instruments is illustrated in Fig. 14Af , 
where the circular fringes produced by a single spectral line are com- 
pared. If a second line were present, it would merely reduce the visibil- 
ity in (a) but would show as a separate set of rings in (6) . As will appear 
later, this fact also permits more exact intercomparisons of wavelength. 

It is important to know how close together two wavelengths may be 
and still be distinguished as separate rings. The ability of any type of 
spectroscope to discriminate wavelengths is expressed as the ratio X/AX, 
where X is the mean wavelength of a barely resolved pair and AX is the 
wavelength difference between the components. This ratio is called the 
chromatic resolving -power of the instrument at that wavelength. In the 
present case, it is convenient to say that the fringes formed by X and 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 



277 




(a) (b) 

Fig. 1421/. Comparison of the types of fringes produced with (a) the Michelson 
interferometer and (6) the Fabry-Perot interferometer with surfaces of reflectance 0.8. 

X -}- AX are just resolved when the intensity contours of the two in a 
particular order lie in the relative positions shown in Fig. 14AT(o). If 
the separation Ad is such as to make the curves cross at the half -intensity 
point, It = 0.5/ , there will be a central dip of 17 per cent in the sum 
of the two, as shown in (b) of the figure. The eye can then easily recog- 
nize the presence of two lines. 




(a) (6) 

Fig. 142V. Intensity contour of two Fabry-Perot fringes that are just resolved, (a) 
Shown separately, (b) Added, to give the observed effect. 

In order to find the AX corresponding to this separation, we note first 
that, in going from the maximum to the halfway point, the phase differ- 
ence in either pattern must change by the amount necessary to make the 
second term in the denominator of Eq. 14/ equal to unity. This requires 
that 



sin" x = 



(1 - r 2 )- 

4r 2 



278 PHYSICAL OPTICS 

If the fringes are reasonably sharp, the change of 5/2 from a multiple of ir 
will be small. Then the sine may be set equal to the angle, and if we 
denote by A 8 the change in going from one maximum to the position 
of the other, we have 

. l/A5\ AS 1 -r 2 n . . 

m l\TJ T^ST (14m) 

Now the relation between an angular change Ad and a phase change A 8 
may be found by differentiating Eq. 14e, setting <t>' = d and n = 1. 

A5 = - ^ sin A6 (14n) 

A 

Furthermore, if the maximum for X 4- AX is to occur at this same angular 
separation Ad, Eq. 14Z requires that 

-2d sin 6 Ad = m AX (14o) 

The combination of Eqs. 14m, 14n, and 14o yields, for the chromatic 
resolving power, 

It thus depends on two quantities, the order m, which may be taken as 
2d/\, and the reflectance r 2 of the surfaces. If the latter is close to 
unity, very large resolving powers are obtained. For example, with 
r 2 = 0.9 the second factor in Eq. 14p becomes 30, and, with a plate sepa- 
ration d of only 1 cm, the resolving power at X5000 becomes 1 .20 X 10 6 . 
The components of a doublet only 0.0042 A wide could be seen as separate. 
14.13. Comparison of Wavelengths with the Interferometer. The 
ratio of the wavelengths of two lines which are not very close together, 
for example, the yellow mercury lines, is sometimes measured in the 
laboratory with the form of interferometer in which one mirror is movable. 
The method is based on observation of the positions of coincidence and 
discordance of the fringes formed by the two wavelengths, a method 
which has already been mentioned in Sec. 13.12. Starting with the two 
mirrors nearly in contact, the ring systems owing to the two wavelengths 
practically coincide. As d is increased, they gradually separate, and the 
maximum discordance occurs when the rings of one set are halfway 
between those of the other set. Confining our attention to the rings at 
the center (cos d = 1), we may write from Eq. 14Z 

2di = rtixk = (m, 4- £)X' (Uq) 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 

where, of course, X > X'. From this, 



279 



and 



m,(X - X') = ^ (X - X') - I' 
\ _ \' — ^ — ^ 



if the difference between X and X' is small. On displacing the mirror 
still farther, the rings will presently coincide and then separate out again. 
At the next discordance 



2d 2 = m 2 X = (m 2 + U)X' 
Subtracting Eq. I4g from Eq. 14r, we obtain 

2(d 2 — di) = (ra 2 — mi)X = (m 2 — mi)X' + X' 
whence, assuming X approximately equal to X', we find 

X 2 



(Mr) 



X - X' - 



2(d 2 - dx) 



(14s) 



We can determine d 2 — di either directly from the scale or by counting 
the number of fringes of the known wavelength X between discordances. 




Ej 



E t 



4 



8 



Fig. 140. Mechanical details of a Fabry-Perot etalon, showing spacer ring, adjusting 
screws, and springs. 

For the most accurate work, the above method is replaced by one in 
which the fringe systems of the lines are photographed simultaneously 
with a fixed separation d of the plates. For this purpose the plates are 
held rigidly in place by quartz or invar spacers. A pair of Fabry-Perot 
plates thus mounted is called an etalon (Fig. 140). The etalon can be 
used to determine accurately the relative wavelengths of several spectral 
lines from a single photographic exposure. If it were mounted with a 
lens as in Fig. 14/C, the light containing several wavelengths, the fringe 
systems of the various wavelengths would be concentric with and 
would be confused with each other. However, they can be separated 



280 



PHYSICAL OPTICS 




Fig. 14P. Fabry-Perot etalon and prism arrangement for separating the ring systems 
produced by different lines. 

by inserting a prism between the etalon and the lens L. The experi- 
mental arrangement is then similar to that shown in Fig. 14P. A photo- 
graph of the visible spectrum of mercury taken in this way is shown in 
the upper part of Fig. 14Q. It will be seen that the fringes of the green 



4358 



X5461 5770-90 




BLUE GREEN YELLOW 

Fig. 14Q. Interference rings of the visible mercury spectrum taken with the Fabry- 
Perot etalon as shown in Fig. 14P. 

and yellow lines still overlap. To overcome this, it is merely necessary 
to use an illuminated slit (MN of Fig. 14P) of the proper width as the 
source. When the interferometer is in a collimated beam of parallel 
light, as it is here, each point on the extended source corresponds to a 
given point in the ring system. Therefore only vertical sections of the 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 281 

ring system are obtained, as shown in the lower part of Fig. 14Q, and 
these no longer overlap. When the spectrum is very rich in lines, as in 
Fig. IAR, the source slit must be made rather narrow. In this photo- 
graph only sections of the upper half of the fringe systems appear. 
Measurements of the radii of the rings in a photograph of this type 
permit very accurate comparisons of wavelengths. The determination 
of the correct values of m in the different systems and of the exact value 



— mm ■*, 


.**» *>» 


m > 


• - 1 


3 Z ""-" "~ 




■ i 

■ 1 


Zz z Z **s* - 


yj+ 




™ "-■ — - .. J**~ 




■ 1 


': ' "---- -*-. 




m a 


■ i 

« ■ 


A A A A A 




XX c 




A X 



Fig. 14/2. Interference patterns of the lanthanum spectrum taken with a Fabry-Perot 
etalon. d = 5 mm. (After Anderson.) 

of rf is a rather involved process which we shall not discuss here.* By 
this method the wavelengths of several hundred lines from the iron arc 
have been measured relative to the red cadmium line within an accuracy 
of a few ten-thousandths of an angstrom. 

14.14. Study of Hyperfine Structure and of Line Shape. Because of 
its bearing on the properties of atomic nuclei, the investigation of hyper- 
fine structure with the Fabry-Perot interferometer has become of con- 
siderable importance in modern research. Occasionally it will be found 
that a line which appears sharp and single in an ordinary spectroscope 
will yield ring systems consisting of two or more sets. Examples are 
found in the lines marked A r in the lanthanum spectrum (Fig. 14/?). 
Those marked A are sharp to a greater or less extent. These multiple 
ring systems arise from the fact that the line is actually a group of lines 
of wavelengths very close together, differing by perhaps a few hundredths 
of an angstrom. If d is sufficiently large, these will be separated, so 
that in each order m we obtain effectively a short spectrum very power- 
fully resolved. Any given fringe of a wavelength X t is formed at such an 
angle that 

2d cos 6 \ = raXi (14^) 

The next fringe farther out for this same wavelength has 

2d cos 2 = (m - 1)\! (].4 W ) 

* See W. E. Williams, "Applications of Interferometiy," 1st ed., pp. 83-88, Methuen 
ft Co., Ltd., London, 1930, for a description of this method. 



282 PHYSICAL OPTICS 

Suppose now that Xi has a component line X2 which is very near Xi, so 
that we may write X2 = Xi — AX. Suppose also that AX is such that 
this component, in order m, falls on the order m — 1 of Xi. Then 

2d cos 6 2 = m(Xi - AX) (14y) 

Equating the right-hand members of Eqs. 14u and 14y, 

Xi = raAX 

Substituting the value of m from Eq. 14/ and solving for AX, 

AX = , Xl2 . ~ ^ (Uw) 

2d cos 0i 2d 

if is nearly zero. This is the wavelength interval in a given order when 
the fringe of the same wavelength in the next higher order is reached. 
We see that it is constant, independent of m. Knowing d and X (approxi- 
mately), the wavelength difference of component lines lying in this small 
range may be evaluated.* 

The equation for the separation of orders becomes still simpler when 
expressed in terms of frequency. Since the frequencies of light are 
awkwardly large numbers, spectroscopists commonly use an equivalent 
quantity called the wave number. This is the number of waves per 
centimeter path in vacuum, and varies from roughly 15,000 to 25,000 cm -1 
in going from red to violet. Denoting wave number by a, we have 

1 k 

To find the wave-number difference A<r corresponding to the AX in Eq. 14w, 
we may differentiate the above equation to obtain 

A(T = — -T-77 



X 



Substitution in Eq. 14i« then yields 



*° = - S (14l) 

Hence, if d is expressed in centimeters, \/2d gives the wave-number 
difference, which is seen to be independent of the order (neglecting the 
variation of 6) and of wavelength as well. 

The study of the width and shape of individual spectrum lines, even 
though they may have no hyperfine structure, is of interest because it can 

* For a good account of the methods, see K. W. Meissner, J. Opt. Soc. Am., 31, 405, 
1941. 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 283 

give us information as to the conditions of temperature, pressure, etc., 
in the light source. If the interferometer has a high resolving power, the 
fringes will have a contour corresponding closely to that of the line itself. 
The small width which is inherent in the instrument can be determined 
by observations with an extremely small etalon spacer, and appropriate 
corrections made. 

The difficult adjustment of the Fabry-Perot interferometer lies in the 
attainment of accurate parallelism of the silvered surfaces. This opera- 
tion is usually accomplished by the use of screws and springs, which 
hold the plates against the spacer rings shown in Fig. 140. A brass 
ring A with three quartz or invar pins constitutes the spacer. A source 
of light such as a mercury arc is set up with a sheet of ground glass G on 
one side of the etalon, and then viewed from the opposite side as shown 
at E. With the eye focused for infinity, a system of rings will be 
seen with the reflected image of the pupil of the eye at its center. As the 
eye is moved up and down or from side to side, the ring system will also 
move along with the image of the eye. If the rings on moving up 
expand in size, the plates are farther apart at the top than at the bottom. 
Tightening the top screw will then depress the corresponding separator 
pin enough to produce the required change in alignment. When the 
plates are properly adjusted, and if they are exactly plane, the rings will 
remain the same size as the eye is moved to any point of the field of view. 

Sometimes it is convenient to place the etalon in front of the slit of a 
spectrograph rather than in front of the prism. In such cases the light 
incident on the etalon need not be parallel. A lens must, however, 
follow the etalon, and this must always be set with the slit at its focal 
plane. This lens selects parallel rays from the etalon and focuses inter- 
ference rings on the slit. Both these methods are used in practice. 

14.16. Other Interference Spectroscopes. When the light is mono- 
chromatic, or nearly so, it is not necessary that the material between the 
highly reflecting surfaces be air. A single accurately plane-parallel glass 
plate having its surfaces lightly silvered will function as a Fabry-Perot 
etalon. The use of two such plates with thicknesses in the ratio of 
whole numbers will result in the suppression of several of the maxima 
produced by the thicker plate, since any light getting through the system 
at a particular angle must satisfy Eq. 14Z for both plates. This arrange- 
ment, known as the compound interferometer, gives the resolving power 
of the thicker plate and the free wavelength range (Eq. 14u>) of the thinner 
one. 

The spacing of the fringes of equal inclination becomes extremely small 
when departs much from 0°. It opens out again, however, near grazing 
incidence. The Lummer-Gehrcke plate makes use of the first few maxima 
near 6 = 90°. In order to get an appreciable amount of light to enter 



284 PHYSICAL OPTICS 

the plate, it is necessary to introduce it by a total-reflection prism 
cemented on one end. It then undergoes multiple internal reflections 
very near the critical angle, and the beams emerging at a grazing angle 
are brought to interference by a lens. High reflectance and resolving 
power are thus obtained with unsilvered surfaces. 

Because of its flexibility, the Fabry-Perot interferometer has for 
research purposes largely replaced such instruments having a fixed 
spacing of the surfaces. For special purposes, however, they may be 
valuable.* 

14.16. Channeled Spectra. Interference Filter. Our discussion of 
the Fabry-Perot interferometer was concerned primarily with the depend- 
ence of the intensity on plate separation and on angle for a single wave- 
length, or perhaps for two or more wavelengths close together. If the 
instrument is placed in a parallel beam of white light, interference will 
also occur for all the monochromatic components of such light, but 
this will not manifest itself until the transmitted beam is dispersed by an 
auxiliary spectroscope. One then observes a series of bright fringes in 
the spectrum, each formed by a wavelength somewhat different from the 
next. The maxima will occur, according to Eq. 14/, at wavelengths 
given by 

2d cos 6 ,, . N 

\ = — - — (14y) 

m 

where m is any whole number. If d is a separation of a few millimeters, 
there will be very many narrow fringes (more than 12,000 through the 
visible spectrum when d = 5 mm), and high dispersion is necessary in 
order to separate them. Such fringes are referred to as a channeled 
spectrum, or as Edser-Butler bands, and have been used, for example, in 
the calibration of spectroscopes for the infrared and in accurate measure- 
ments of wavelengths of the absorption lines in the solar spectrum. 

An application of these fringes having considerable practical impor- 
tance uses the situation where d is extremely small, so that only one or 
two maxima occur within the visible range of wavelengths. With white 
light incident, only one or two narrow bands of wavelength will then be 
transmitted, the rest of the light being reflected. The pair of semitrans- 
parent metallic films thus can act as a filter passing nearly monochromatic 
light. The curves of transmitted energy against wavelength resemble 
those of Fig. 14/f, since according to Eq. 14e the phase difference 5 is 
inversely proportional to wavelength for a given separation d. 

In order that the maxima shall be widely separated, it is necessary 
that m be a small number. This is attained only by having the reflecting 

* For a more detailed description of these and other similar instruments, see A. C. 
Candler, "Modern Interferometers," Hilger and Watts, London, 1951. 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 



285 



surfaces very close together. If one wishes to have the maximum for 
m = 2 occur at a given wavelength X, the metal films would have to be 
a distance X apart. The maximum m = 1 will then appear at a wave- 
length of 2X. Such minute separations can be attained, however, with 
modern techniques of evaporation in vacuum. A semitransparent metal 
film is first evaporated on a plate of glass. Next, a thin layer of some 
dielectric material such as cryolite (3NaF-AlF 3 ) is evaporated on top of 
this, and then the dielectric layer is in turn coated with another similar 
film of metal. Finally another plate of glass is 
placed over the films for mechanical protection. 
The completed filter then has the cross section 
shown schematically in Fig. 14$, where the thick- 
ness of the films is greatly exaggerated relative 
to that of the glass plates. Since the path diff- 
erence is now in the dielectric of index n, the 
wavelengths of maximum transmission for nor- 
mal incidence are given by 



Evaporated 
metal films 

1 



X = 



2nd 
m 



(142) 




Evaporated layer 
of transparent material 
Fig. 14S. Cross section of 
an interference filter. 



If there are two maxima in the visible spectrum, 

one of them can easily be eliminated by using 

colored glass for the protecting cover plate. 

Interference filters are now made which transmit 

a band of wavelengths of width (at half transmission) only 15 A, with 

the maximum lying at any desired wavelength. The transmission at the 

maximum can be as high as 45 per cent. It is very difficult to obtain 

combinations of colored glass or gelatin filters which will accomplish this 

purpose. Furthermore, since the interference filter reflects rather than 

absorbs the unwanted wavelengths, there is no trouble with its overheating. 



PROBLEMS 

1. Fringes of equal thickness are often used to compare the lengths of standard end 
gauges, which consist of cylindrical pieces of steel with ends that are accurately flat 
and parallel. Suppose that two of these which are nominally of equal length are 
placed on an optical flat, and another glass flat laid across the top. If sodium fringes 
are then formed between the latter and the top surfaces of the gauges, it is found 
that there are eight fringes per centimeter. The points of contact of the flat with 
the two gauges are 5 cm apart. Find the difference in length of the gauges. 

2. In an experiment with Newton's rings, the diameters of the sixth and twentieth 
bright rings formed by the green mercury line are measured to be 1.76 and 3.22 mm, 
respectively. Calculate the radius of curvature of the convex surface. Arts. 23.78 cm. 

3. Three spherical surfaces of large radius are to be compared by observing New- 
ton's rings when they are placed together in pairs. The diameters of the sixteenth 



286 PHYSICAL OPTICS 

bright ring in the three possible combinations are found to be 16.0, 20.8, and 12.8 mm, 
with light of wavelength 5000 A. Find the three radii of curvature. 

4. A nonreflecting layer is to be deposited on the surface of a lens having n = 
1.780. Assuming that the coating material has an index of 1.334, what would be the 
necessary thickness for zero reflection at 5500 A? What would be the reflectance 
of the layer at 6500 A? The reflectance of either surface may be taken as 2.05 per 
cent in both cases. Ans. 1.031 X 10~ 5 cm; 0.47%. 

6. Using vector diagrams, find the resultant amplitude and intensity in the inter- 
ference pattern from a Fabry-Perot interferometer having a reflectance of 70 per cent, 
when the phase difference is (a) 0, (b) w/S, (c) 7r/4. Carry the vectors far enough to 
obtain the relative intensities to within 2 per cent. 

6. The plates of a Fabry-Perot interferometer have a reflectance of 0.85. Calcu- 
late the minimum separation of the plates required to resolve the components of the 
H line of hydrogen, which is a doublet of wavelength difference 0.136 A. 

Ans. 0.82 mm. 

7. Prove that the fringes in a channeled spectrum produced by means of an air film 
are separated by equal wave-number intervals. 

8. The method of coincidences of Fabry-Perot rings is used to compare two wave- 
lengths, one of which is exactly 4800 A and the other slightly greater. If coincidences 
occur at plate separations of 1.90, 2.50, and 3.10 mm, find the unknown wavelength. 

Ana. 4801.92 A 

9. If in taking the Fabry-Perot spectrogram of Fig. 1472 the plate spacing had been 
exactly 5 mm, what would be the wavelength separation of orders for a line at 5000 A? 
What would be the linear diameter of the tenth order from the center if the focal 
length of the camera lens were 1 m? 

10. A channeled spectrum is formed by placing a microscope slide 1 mm thick in 
front of the slit of a spectroscope illuminated by white light. If the glass has no = 
1.546, find the separation of maxima in angstroms near the sodium D lines. 

Ans. 1.12 A. 

11. The Fabry-Perot rings for a line at 4649 A, which has hyperfine structure, are 
photographed with an etalon spacer of 9.0 mm. The difference in the squares of the 
diameters of the rings formed by the strongest component is constant and equal to 
28.65 mm 2 . Two weaker components lie adjacent to the strong one, on the side 
opposite the center of the ring system. The squares of their diameters differ by 
7.95 and 14.43 mm 2 , respectively, from those for the strong component. Find their 
respective separations, in angstroms, from the main line, including the sign of the 
shift. 

12. Design an interference filter using a layer of cryolite (n = 1.35) to separate two 
semitransparent silver layers. The filter is to transmit the yellow mercury lines 
(those of longest wavelength in Table 21-1) with a maximum of intensity and reduce 
the green line to less than 1 per cent. Find (a) the thickness of the cryolite, neglect- 
ing phase changes at the silver surfaces, and (b) the minimum reflectance required 
for the silver layers. Ans. (a) 2.141 X lO" 5 cm. (6) 96.4%. 

13. Prove that in the fringe system formed by a Fabry-Perot interferometer the 
contrast, or the ratio of the intensity of maxima to the intensity midway between 
maxima, is given by (1 4- r 2 ) 2 /(l — r 2 ) 2 . 

14. Show that the second factor in Eq. 14p, namely irr/(l — r 9 ), represents the 
ratio of the separation of the fringes to their width at half intensity. (Hint: To find 5 
for half intensity, put Ir/h — 0.5 in Eq. 14j.) 

16. A glass Lummer-Cichrcke plate is 10 cm long and 6 mm thick. If the refrac- 
tive index for light of the blue cadmium line, X4799.91, is 1.632, find the order of 



INTERFERENCE INVOLVING MULTIPLE REFLECTIONS 287 

interference at grazing emergence. Find also the number of beams brought to 
interference. 

16. Fringes of equal inclination are formed with a plane-parallel glass plate of 
index 1.50 and 2 mm thick. How many fringes are formed in the entire range from 
normal incidence to grazing incidence? Calculate the maximum and minimum 
values of their angular spacing. Take X = 6000 A. 

Ans. 2546 fringes. 1.22° near 4> = 0. 0.0225° at <f> = 49°12'. 

17. The plates of a Fabry-Perot interferometer are silvered to such a density that 
each reflects 90 per cent, transmits 4 per cent, and absorbs 6 per cent at a given wave- 
length. Find the intensity at the maximum of the rings, as compared with the value 
it would have were there no absorption. 



CHAPTER 15 

FRAUNHOFER DIFFRACTION BY A SINGLE OPENING 



When a beam of light passes through a narrow slit, it spreads out to a 
certain extent into the region of the geometrical shadow. This effect, 
already noted and illustrated at the beginning of Chaps. 1 and 13, is one 
of the simplest examples of diffraction, i.e., of the failure of light to travel 
in straight lines. It can be satisfactorily explained only by assuming a 
wave character for light, and in this chapter we shall investigate quantita- 
tively the diffraction pattern, or distribution of intensity of the light behind 
the aperture, using the principles of wave motion already discussed. 

16.1. Fresnel and Fraunhofer Diffraction. Diffraction phenomena 
are conveniently divided into two general classes, (1) those in which the 
source of light and the screen on which the pattern is observed are effec- 
tively at infinite distances from the aperture causing the diffraction, and 
(2) those in which either the source or the screen, or both, are at finite 
distances from the aperture. The phenomena coming under class (1) 
are called, for historical reasons, Fraunhofer diffraction, and those coming 
under class (2) Fresnel diffraction. Fraunhofer diffraction is much sim- 
pler to treat theoretically. It is easily observed in practice by rendering 
the light from a source parallel with a lens, and focusing it on a screen 
with another lens placed behind the aperture, an arrangement which 
effectively removes the source and screen to infinity. In the observation 
of Fresnel diffraction, on the other hand, no lenses are necessary, but 
here the wave fronts are divergent instead of plane, and the theoretical 
treatment is consequently more complex. Only Fraunhofer diffraction 
will be considered in this chapter. 

15.2. Diffraction by a Single Slit. A slit is a rectangular aperture of 
length large compared to its breadth. Consider a slit S to be set up as 
in Fig. 15 A, with its long dimension perpendicular to the plane of the 
page, and to be illuminated by parallel monochromatic light from the 
narrow slit S', at the principal focus of the lens L\. The light focused 
by another lens L 2 on a screen or photographic plate P at its principal 
focus will form a diffraction pattern, as indicated schematically. Figure 
\5B(b) and (c) shows two actual photographs, taken with different expo- 
sure times, of such a pattern, using violet light of wavelength 4358 A. 

288 



FRAUNHOFER DIFFRACTION BY A SINGLE OPENING 



289 





Source 
slit 



Fig. 15/1. Experimental arrangement for obtaining the diffraction pattern of a single 
slit. Fraunhofer diffraction. 

The distance S'L t was 25 cm, and L 2 P was 100 cm. The width of the 
slit S was 0.090 mm, and of S', 0.10 mm. If S' was widened to more 
than about 0.3 mm, the details of the pattern began to be lost. On the 
original plate, the half width d of the central maximum was 4.84 mm. It is 
important to notice that the width of the central maximum is twice as 
great as that of the fainter side maxima. That this effect comes under 




Fig. 15B. Photographs of the single-slit diffraction pattern. 

the heading of diffraction as previously defined is clear when we note 
that the strip drawn in Fig. 15/? (a) is the width of the geometrical 
image of the slit S', or practically that which would be obtained by 
removing the second slit and using the whole aperture of the lens. This 
pattern can easily be observed by ruling a single transparent line on a 
photographic plate and using it in front of the eye as explained in Sec. 13.2. 






290 



PHYSICAL OPTICS 



The explanation of the single-slit pattern lies in the interference ol 
the Huygens secondary wavelets which can be thought of as sent out 
from every point on the wave front at the instant that it occupies the 
plane of the slit. To a first approximation, one may consider these wave- 
lets to be uniform spherical waves, the emission of which stops abruptly 
at the edges of the slit. The results obtained in this way, although 
they give a fairly iccurate account of the observed facts, are subject to 




Fig. 15C. Geometrical construction for investigating the intensity in the single-slit 
diffraction pattern. 

certain modifications in the light of the more rigorous theory to be men- 
tioned in Sec. 18. L7. 

Figure 15C represents a section of a slit of width b, illuminated by 
parallel light from the left. Let ds be an element of width of the wave 
front in the plane of the slit, at a distance s from the center 0, which we 
shall call the origin. The parts of each secondary wave which travel 
normal to the plane of the slit will be focused at Po, while those which 
travel at any angle 6 will reach P. Considering first the wavelet emitted 
by the element ds situated at the origin, its amplitude will be directly 
proportional to the length ds, and inversely proportional to the distance 
x. At P it will produce an infinitesimal displacement which, according 
to Eq. 11; for a spherical wave, may be expressed as 

dyo = sin (ut — kx) 

■v 

As the position of ds is varied, the displacement it produces will vary in 
phase because of the different path length to P. When it is at a distance 



FRAUNHOFER DIFFRACTION IJY A SINGLE 9PENING 291 

s below the origin, the contribution will be 

dy, = sin [ut — k(x + A)] 

= sin (o)t — kx — ks sin 6) (15a) 

x 

We now wish to sum the effects of all elements from, one edge of the slit 
to the other. This may be done by integrating Eq. 15a from s = — 6/2 
to 6/2. The simplest way* is to integrate the contributions from pairs 
of elements symmetrically placed at s and — s, each contribution being 

dy = dy-, + dy, 

= [sin (a)t — kx — ks sin 6) + sin ut — kx + ks sin 6)] 

By the identity sin a + sin = 2 cos ?(a — /S) sin %(a + 0), we have 

dy = [2 cos (ks sin 0) sin {oil — kx)] 

which must be integrated from s = to 6/2. In doing so, x may be 
regarded as constant, insofar as it affects the amplitude. Thus 

2a . f b/2 

y = — sin (oit — kx) I cos (ks sin 0) ds 
£ Jo 

sin (fcssin 0) lb 



= 2a [si 



k sin 6 . o 
a6 sin (£&& sin 6) 
x \kb sin 8 



sin (ul — kx) 
sin (at — kx) (156) 



The resultant vibration will therefore be a simple harmonic one, the 
amplitude of which varies with the position of P, since the latter is deter- 
mined by 6. We may represent its amplitude as 

a a si n /, . x 

A = A —^ (15c) 

where /3 = £&6 sin 6 = (x6 sin 6)/\ and Ao = ab/x. The quantity 
is a convenient variable, which signifies one-half the phase difference 
between the contributions coming from opposite edges of the slit. The 
intensity on the screen is then 

I~A> = A > S -^ (15d) 

* The method of complex amplitudes (Sec. 14.8) starts with (ab/x)fe ik,BlD B ds, and 
yields the real amplitude upon multiplication of the result by its complex conjugate. 
No simplification results from using the method here. 



292 



PHYSICAL OPTICS 



If the light, instead of being incident on the slit perpendicular to its 

plane, makes an angle i, a little consideration will show that it is merely 

necessary to replace the above expression for /3 by the more general 

expression 

7r6(sin i + sin 6) /-.- \ 

/? = — i (loe) 

15.3. Further Investigation of the Single-slit Diffraction Pattern. In 
Fig. 15 D (a) graphs are shown of Eq. 15c for the amplitude (dotted curve) 



litude 




Fig. 15D. Amplitude and intensity contours for Fraunhofer diffraction of a single slit, 
showing positions of maxima and minima. 



and Eq. 15d for the intensity, taking the constant A in each case as 
unity. The intensity curve will be seen to have the form required by 
the experimental result in Fig. 15/3. The maximum intensity of the 
strong central band comes at the point P of Fig. 15C, where evidently 
all the secondary wavelets will arrive in phase because the path differ- 
ence A = 0. For this point /3 = 0, and although the quotient (sin /3)//3 
becomes indeterminate for /S = 0, it will be remembered that sin /3 
approaches /3 for small angles, and is equal to it when vanishes. Hence 
for jS = 0, (sin /3)//3 = 1. We now see the significance of the constant 
Ao. Since for /3 = 0, A = Aq, it represents the amplitude when all the 
wavelets arrive in phase. Ao 2 is then the value of the maximum inten- 
sity, at the center of the pattern. From this principal maximum the 




FRAUNHOFER DIFFRACTION BY A SINGLE OPENING 293 

intensity falls to zero at = ±x, then passes through several secondary 
maxima, with equally spaced points of zero intensity at /3 = ±ir, ±2w, 
±3t, . . . , or in general = mac. The secondary maxima do not fall 
halfway between these points, but are displaced toward the center of 
the pattern by an amount which 
decreases with increasing m. The 
exact values of for these max- 
ima can be found by differentiat- 
ing Eq. 15c with respect to and 
equating to zero. This yields the 
condition 

tan = 

The values of satisfying this re- 
lation are easily found graphically Intensity 
as the intersections of the curve Fig. 152?. Angle of the first minimum of 
V « tan and the straight line the sin g le - s,it diffraction pattern. 
y = 0. In Fig. 15D(6) these points of intersection lie directly below the 
corresponding secondary maxima. 

The intensities of the secondary maxima may be calculated to a very 
close approximation by finding the values of (sin 2 0)/0 2 at the halfway 
positions, i.e., where = 3tt/2, &r/2, 7tt/2, .... This gives 4/(9tt 2 ), 

4/(25tt 2 ), 4/(49ir 2 ), . . . , or ~^, ^L, -L . . . # of the intensity of 

the principal maximum. Reference to Table 15-1 ahead, where the exact 
values of the maxima are given, will show that the approximation is very 
good. The first secondary maximum, where the deviation is largest, is 
actually 4.72 per cent of the central intensity, whereas the above figure 
of 1/22.2 corresponds to 4.50 per cent. 

A very clear idea of the origin of the single-slit pattern is obtained 
by the following simple treatment. Consider the light from the slit of 
Fig. 1527 coming to the point Pi on the screen, this point being just one 
wavelength farther from the upper edge of the slit than from the lower. 
The secondary wavelet from the point in the slit adjacent to the upper 
edge will travel approximately A/2 farther than that from the point at 
the center, and so these two will produce vibrations with a phase differ- 
ence of t and will give a resultant displacement of zero at P x . Similarly 
the wavelet from the next point below the upper edge will cancel that 
from the next point below the center, and we may continue this pairing 
off to include all points in the wave front, so that the resultant effect 
at Pi is zero. At P 3 the path difference is 2A, and if we divide the slit 
into four parts, the pairing of points again gives zero resultant, since 
the parts cancel in pairs. For the point P 2 , on the other hand, the path 



294 PHYSICAL OPTICS 

difference is 3X/2, and we may divide the slit into thirds, two of which 
will cancel, leaving one third to account for the intensity at this point. 
The resultant amplitude at P2 is, of course, not even approximately 
one-third that at Po, because the phases of the wavelets from the remain- 
ing third are not by any means equal. 

The above method, though instructive, is not exact if the screen is 
at a finite distance from the slit. As Fig. 15E is drawn, the shorter 
broken line is drawn to cut off equal distances on the rays to Pi. It will 
be seen from this that the path difference to Pi between the light coming 
from the upper edge and that from the center is slightly greater than 
X/2, and that between the center and lower edge slightly less than X/2. 
Hence the resultant intensity will not be zero at Pi and P 3 , but it will 
be more nearly so the greater the distance between slit and screen, or 
the narrower the slit. This corresponds to the transition from Fresnel 
diffraction to Fraunhofer diffraction. Obviously, with the relative 
dimensions shown in the figure, the geometrical shadow of the slit would 
considerably widen the central maximum as drawn. Just as was true 
with Young's experiment (Sec. 13.3), when the screen is at infinity, the 
relations become simpler. Then the two angles 0i and 6[ in Fig. \bE 
become exactly equal (i.e., the two broken lines are perpendicular to each 
other), and X = b sin 0i for the first minimum corresponding to /? = x. 

In practice 6\ is usually a very small angle, so we may put the sine 
equal to the angle. Then 

*-\ 

a relation which shows at once how the dimensions of the pattern vary 
with X and b. The linear width of the pattern on a screen will be pro- 
portional to the slit-screen distance, which is the focal length / of a lens 
placed close to the slit. The linear distance d between successive minima 
corresponding to the angular separation 6\ = X/6 is thus 

a b 

The width of the pattern increases in proportion to the wavelength, so 
that for red light it is roughly twice as wide as for violet light, the slit 
width, etc., being the same. If white light is used, the central maximum 
is white in the middle, but is reddish on its outer edge, shading into a 
purple and other impure colors farther out. 

The angular width of the pattern for a given wavelength is inversely 
proportional to the slit width 6, so that as b is made larger, the pattern 
shrinks rapidly to a smaller scale. In photographing Fig. 15J3, if the 
slit S had been 9 mm wide, the whole visible pattern (of five maxima) 



FRAUNHOFER DIFFRACTION BY A SINGLE OPENING 295 

would be included in a width of 0.24 mm on the original plate instead of 
2.4 cm. This fact, that when the width of the aperture is large compared 
to a wavelength the diffraction is practically negligible, led the early 
investigators to conclude that light travels in straight lines and that it 
could not be a wave motion. Sound waves, whose lengths are measured 
in feet, will evidently be diffracted through large angles in passing through 
an aperture of ordinary size, such as an open window. 

15.4. Graphical Treatment of Amplitudes. The Vibration Curve. 
The addition of the amplitude contributions from all the secondary wave- 
lets originating in the slit may be carried out by a graphical method based 
on the vector addition of amplitudes discussed in Sec. 12.2. It will be 
worth while to consider this method in some detail, because it may be 
applied to advantage in other more complicated cases to be treated in 
later chapters, and because it gives a very clear physical picture of the 
origin of the diffraction pattern. Let us divide the width of the slit into a 
fairly large number of equal parts, say 9. The amplitude r contributed 
at a point on the screen by any one of these parts will be the same, since 
they are of equal width. The phases of these contributions will differ, 
however, for any point except that lying on the axis, i.e., on the normal 
to the slit at its center (P , Fig. 15C). For a point off the axis, each of 
the 9 segments will contribute vibrations differing in phase, because the 
segments are at different average distances from the point. Further- 
more the difference in phase 8 between the contributions from adjacent 
segments will be constant, since each element is on the average the same 
amount farther away (or nearer) than its neighbor. 

Now, using the fact that the resultant amplitude and phase may be 
found by the vector addition of the individual amplitudes making angles 
with each other equal to the phase difference, a vector diagram like that 
shown in Fig. 15F(6) may be drawn. Each of the 9 equal amplitudes a 
is inclined at an angle 8 with the preceding one, and their vector sum A 
is the resultant amplitude required. Now suppose that instead of 
dividing the slit into 9 elements, we had divided it into many thousand 
or, in the limit, an infinite number of equal elements. The vectors a 
would become shorter, but at the same time 8 would decrease in the 
same proportion, so that in the limit our vector diagram would approach 
the arc of a circle, shown as in (6'). The resultant amplitude A is still 
the same and equal to the length of the chord of this arc. Such a con- 
tinuous curve, representing the addition of infinitesimal amplitudes, we 
shall refer to as a vibration curve. 

To show that this method is in agreement with our previous result, 
we note that the length of the arc is just the amplitude A obtained when 
all of the component vibrations are in phase, as in (a) of the figure. 
Introducing a phase difference between the components does not alter 



296 



PHYSICAL OPTICS 



their individual amplitudes or the algebraic sum of these. Hence the 
ratio of the resultant amplitude A at any point in the screen to A , that 
on the axis, is the ratio of the chord to the arc of the circle. Since /3 
stands for half the phase difference from opposite edges of the slit, the angle 
subtended by the arc is just 2/3, because the first and last vectors a will 
have a phase difference of 2/3. In Fig. l5F(b'), the radius of the arc is 




/-v A '-° 

©o 



(c) 



/3=3 7r /4 
(d) 



3--7T 0'5% s 3*/2 0=7*/4 = 2* 

(e) (/) (g) (M U) 



Fig. 15F. Graphical treatment of amplitudes in single-slit diffraction. 



called q, and a perpendicular has been dropped from the center on the 
chord A. From the geometry of the figure, we have 



sin /3 = 



A/2 



A = 2q sin /3 



and hence 



A _ chord _ 2q sin /3 _ sin ft 
~A~ ~ arc " q X 2/3 " /3 



in agreement with Eq. 15c. 

As we go out from the center of the diffraction pattern, the length of 
the arc remains constant and equal to A , but its curvature increases 
owing to the larger phase difference S introduced between the infinitesi- 
mal component vectors a. The vibration curve thus winds up on itself 
as /3 increased. The successive diagrams (a) to (z) in Fig. 15F are 
drawn for the indicated values of at intervals of jr/4, and the corre- 
sponding points are similarly lettered on the intensity diagram. A study 



Qtr 



FRAUNHOFER DIFFRACTION BY A SINGLE OPENING 297 

of these figures will bring out clearly the cause of the variations in inten- 
sity occurring in the single-slit pattern. In particular, one sees that the 
asymmetry of the secondary maxima follows from the fact that the 
radius of the circle is shrinking with increasing 0. Thus A will reach 
its maximum length slightly before the condition represented in Fig. 
15F&). 

15.5. Rectangular Aperture. In the preceding sections the intensity 
function for a slit was derived by summing the effects of the spherical 
wavelets originating from a linear section of the wave front by a plane 




Fig. 15G. Diffraction pattern from a rectangular opening. (After A. Kohler.) 

perpendicular to the length of the slit, i.e., by the plane of the page in 
Fig. 15C. Nothing was said about the contributions from parts of the 
wave front out of this plane. A more thorough mathematical investi- 
gation, involving a double integration over both dimensions of the wave 
front,* shows, however, that the above result is correct when the slit is 
very long compared to its width. The complete treatment gives, for a 
slit of width b and length I, the following expression for the intensity: 

r , „,„ sin 2 /S sin 2 y 

^W-pr'—r- (is/) 

where p = (irb sin 0)/X, as before, and y = (tI sin Q)/\. The angles 6 
and ft are measured from the normal to the aperture at its center, in 
planes through the normal parallel to the sides b and I, respectively. 
The diffraction pattern given by Eq. 15/ when b and I are comparable 
with each other is illustrated in Fig. 15G. The dimensions of the aper- 

* See R. W. Wood, "Physical Optics," 2d ed., pp. 195-202, The Macmillan Com- 
pany, New York, 1921. 



298 PHYSICAL OPTICS 

ture are shown by the white rectangle in the lower left-hand part of the 
figure. The intensity in the pattern is concentrated principally in two 
directions coinciding with the sides of the aperture, and in each of these 
directions it corresponds to the simple pattern for a slit width equal to 
the width of the aperture in that direction. Owing to the inverse pro- 
portionality between the slit width and the scale of the pattern, the 
fringes are more closely spaced in the direction of the longer dimension 
of the aperture. In addition to these patterns there are other faint 
maxima, as shown in the figure. This diffraction pattern may easily be 
observed by illuminating a small rectangular aperture with monochro- 
matic light from a source which is effectively a point, the disposition of 
the lenses and the distance of the source and screen being similar to those 
described for observation of the slit pattern (Sec. 15.2). The cross 
formed by the brightest spots in the photograph is the one always 
observed when a bright street light is seen through a wet umbrella. 

Now for a slit having I very large, the factor (sin 2 7V7 2 in Eq. 15/ is 
zero for all values of fi except extremely small ones. This means that 
the diffraction pattern will be limited to a line on the screen perpendicu- 
lar to the slit and will resemble a section of the central horizontal line of 
bright spots in Fig. 15(7. We do not ordinarily observe such a line pat- 
tern in diffraction by a slit, because its observation requires the use of a 
■point source. In Fig. \bA the primary source was a slit S', with its long 
dimension perpendicular to the page. In this case, each point of the 
source slit forms a line pattern, but these fall adjacent to each other on 
the screen, adding up to give a pattern like Fig. 155. If we were to use 
a slit source with the rectangular aperture of Fig. 15G, the slit being 
parallel to the side I, the result would be the summation of a number of 
such patterns, one above the other, and would be identical with Fig. 15B. 

These considerations can easily be extended to cover the effect of 
widening the primary slit. With a slit of finite width, each line element 
parallel to the length of the slit forms a pattern like Fig. 15J5. The 
resultant pattern is equivalent to a set of such patterns displaced laterally 
with respect to each other. If the slit is too wide, the single-slit pattern 
will therefore be lost. No great change will occur until the patterns 
from the two edges of the slit are displaced about one-fourth of the dis- 
tance from the central maximum to the first minimum. This condition 
will hold when the width of the primary slit subtends an angle of -§■ (X/6) 
at the first lens, as can be seen by reference to Fig. 15H below. 

16.6. Resolving Power with a Rectangular Aperture. By the resolv- 
ing power of an optical instrument we mean its ability to produce separate 
images of objects very close together. Using the laws of geometrical 
optics, a telescope or a microscope is designed to give an image of a 
point source which is as small as possible. However, in the final analysis, 



FRAUNHOFER DIFFRACTION BY A SINGLE OPENING 299 

it is the diffraction pattern that sets a theoretical upper limit to the 
resolving power. We have seen that whenever parallel light passes 
through any aperture, it cannot be focused to a point image, but instead 
gives a diffraction pattern in which the central maximum has a certain 
finite width, inversely proportional to the width of the aperture. The 
images of two objects will evidently not be resolved if their separation 
is much less than the width of the central diffraction maximum. The 
aperture here involved is usually that of the objective lens of the telescope 
or microscope and is therefore circular. Diffraction by a circular aper- 
ture will be considered below in Sec. 15.8, and here we shall treat the 
somewhat simpler case of a rectangular aperture. 



8u 



8k 



Intensity 
Fig. \bH. Diffraction images of two slit sources formed by a rectangular aperture. 

Figure 15// shows two plano-convex lenses (equivalent to a single 
double-convex lens) limited by a rectangular aperture of vertical dimen- 
sion b. Two narrow slit sources S, and S 2 perpendicular to the plane 
of the figure form real images S[ and S' 2 on a screen. Each image con- 
sists of a single-slit diffraction pattern for which the intensity distribution 
is plotted in a vertical direction. The angular separation a of the central 
maxima is equal to the angular separation of the sources, and with the 
value shown in the figure is adequate to give separate images. The 
condition illustrated is that in which each principal maximum falls 
exactly on the second minimum of the adjacent pattern. This is the 
smallest possible value of a which will give zero intensity between the 
two strong maxima in the resultant pattern. The angular separation 
from the center to the second minimum in either pattern then corresponds 
to = 2ir (see Fig. 15/)), or sin 6 ~ 6 = 2X/6 = 20 x . As a is made 
smaller than this, and the two images move closer together, the intensity 
between the maxima will rise, until finally no minimum remains at the 
center. Figure 15/ illustrates this by showing the resultant curve (heavy 
line) for four different values of a. In each case the resultant pattern 
has been obtained by merely adding the intensities due to the separate 
patterns (dotted and light curves), as was done in the case of the Fabry- 
Perot fringes (Sec. 14.12). 



300 PHYSICAL OPTICS 

Inspection of this figure shows that it would be impossible to resolve 
the two images if the maxima were much closer than a = 0i, correspond- 
ing to /3 = ir. At this separation the maximum of one pattern falls 
exactly on the first minimum of the other, so that the intensities of the 
maxima in the resultant pattern are equal to those of the separate 
maxima. The calculations are therefore simpler than for Fabry-Perot 
fringes, where at no point does the intensity actually become zero. To 
find the intensity at the center of the resultant minimum for diffraction 




Fig. 157. Diffraction images of two slit sources, 
resolved, (d) Not resolved. 



(a) and (b) Well resolved. 



(c) Just 



fringes separated by 8 h we note that the curves cross at /S = tr/2 for either 
pattern and 



sin 2 /3 



= - = 0.4053 



the intensity of either relative to the maximum. The sum of the con- 
tributions at this point is therefore 0.8106, which shows that the intensity 
of the resultant pattern drops almost to four-fifths of its maximum value. 
This change of intensity is easily visible to the eye, and in fact a consid- 
erably smaller change could be seen, or at least detected with a sensitive 
intensity-measuring instrument such as a microphotometer. However, 
the depth of the minimum changes very rapidly with separation in this 
region, and in view of the simplicity of the relations in this particular case, 
it was decided by Rayleigh to arbitrarily fix the separation a = di = A/6 
as the criterion for resolution of two diffraction patterns. This quite 
arbitrary choice is known as "Rayleigh's criterion." The angle 0i is 
sometimes called the "resolving power" of the aperture b, although the 



FRAUNHOFER DIFFRACTION BY A SINGLE OPENING 



301 



ability to resolve increases as X becomes smaller. A more appropriate 
designation for 6i is the minimum angle of resolution. 

15.7. Chromatic Resolving Power of a Prism. An example of the use 
of this criterion for the resolving power of a rectangular aperture is 
found in the prism spectroscope, if we assume that the face of the prism 
limits the refracted beam to a rectangular section. Thus, in Fig. 15J, 
the minimum angle A5 between two parallel beams which give rise to 
images on the limit of resolution is such that A5 = di = \/b, where b 




Fig. 15/. Resolving power of a prism. 



is the width of the emerging beam. The two beams giving these images 
differ in wavelength by a small increment AX, which is negative because 
the smaller wavelengths are deviated through greater angles. The wave- 
length increment is more useful than the increment of angle, and is the 
quantity that enters in the chromatic resolving power X/AX (Sec. 14.12). 
To evaluate this for the prism, we first note that, since any optical path 
between two successive positions b' and b of the wave front must be the 
same, we can write 

c + c' = nB (150) 

Here n is the refractive index of the prism for the wavelength X, and B 
the length of the base of the prism. Now, if the wavelength be decreased 
by AX, the optical path through the base of the prism becomes (n + An)B, 
and the emergent wave front must turn through an angle A5 = X/6 in 
order that the image it forms may be just resolved. Since, from the 
figure, A5 = (Ac)/b, this amount of turning increases the length of the 
upper ray by Ac = X. It is immaterial whether we measure Ac along the 
rays X or X + AX, because only a difference of the second order is involved. 
Then we have 

c + c' + X = (ft + An)B 



302 



PHYSICAL OPTICS 



and, subtracting Eq. I5g, 



X = B An 



The desired result is now obtained by dividing by AX and substituting 
the derivative dn/d\ for the ratio of small increments. 



X R dn 

AX ~ d\ 



(15h) 



It is not difficult to show (see Prob. 1) that this expression also equals 
the product of the angular dispersion and the width b of the emergent 
beam. Furthermore, we find that Eq. 15/i can still be applied when the 
beam does not fill the prism, in which case B must be the difference in the 
extreme paths through the prism, and when there are two or more prisms 
in tandem, when B is the sum of the bases. 

15.8. Circular Aperture. The diffraction pattern formed by plane 
waves from a point source passing through a circular aperture is of con- 
siderable importance as applied to the resolving power of telescopes and 
other optical instruments. Unfortunately it is also a problem of con- 
siderable difficulty, since it requires a double integration over the surface 

Table 15-1 





Circular aperture 


Single slit 


Ring 


m 


'max 


Itotal 


in 


' max 






1.220 

1.635 

2.233 

2.679 

3 . 238 


1 

0.01750 
0.00416 


1 

0.084 
0.033 




1.000 
1.430 
2.000 
2 . 459 


l 








0.0472 






Third bright 


0.0165 


Third dark 


3.000 






3.699 0.00160 

4.241 

4.710 0.00078 

5.243 


0.018 
0.011 


3.471 
4.000 

4.477 


0.0083 






Fifth bright 


0.0050 


Fifth dark 


5.000 

















of the aperture similar to that mentioned in Sec. 15.5 for a rectangular 
aperture. The problem was first solved by Airy* in 1835, and the solu- 
tion is obtained in terms of Bessel functions of order unity. These 
must be calculated from series expansions, and the most convenient way 
to express the results for our purpose will be to quote the actual figures 
obtained in this way (Table 15-1). 

* Sir George Airy (1801-1892). Astronomer Royal of England from 1835 to 1881. 
Also known for his work on the aberration of light (Sec. 19.13). For details of the 
solution here referred to, see T. Preston, "Theory of Light," 5th ed., pp. 324-327, 
Macmillan & Co., Ltd., London, 1928. 



FRAUNHOFER DIFFRACTION BY A SINGLE OPENING 303 

The diffraction pattern as illustrated in Fig. \5K(a) consists of a 
bright central disk, known as Airy's disk, surrounded by a number of 
fainter rings. Neither the disk nor the rings are sharply limited but 
shade gradually off at the edges, being separated by circles of zero inten- 
sity. The intensity distribution is very much the same as that which 
would be obtained with the single-slit pattern illustrated in Fig. 15E 
by rotating it about an axis in the direction of the light and passing 
through the principal maximum. The dimensions of the pattern are, 




Fig. 15K. Photographs of diffraction images of point sources taken with a circular 
aperture, (a) One source. (6) Two sources just resolved, (c) Two sources com- 
pletely resolved. 

however, appreciably different from those in a single-slit pattern for a 
slit of width equal to the diameter of the circular aperture. For the 
single-slit pattern, the angular separation 6 of the minima from the 
center was found in Sec. 15.3 to be given by sin 6 a± d = m\/b, where m 
is any whole number, starting with unity. The dark circles separating 
the bright ones in the pattern from a circular aperture may be expressed 
by a similar formula, if 6 is now the angular semidiameter of the circle, 
but in this case the numbers m are not integers. Their numerical values 
as calculated by Lommel* are given in Table 15-1. This table also 
includes the values of m for the maxima of the bright rings, and data on 
their intensities. The column headed /^ gives the relative intensities 
of the maxima, while that headed T^tai is the total amount of light in the 
ring, relative to that of the central disk. For comparison, the values 
of in and /„,, for the straight bands of the single-slit pattern are also 
included. 

* E. V. Lommel, Abhandl. Bayer Akad. Wiss., 15, 531, 1886. 



304 PHYSICAL OPTICS 

15.9. Resolving Power of a Telescope. To give an idea of the linear 

size of the above diffraction pattern, let us calculate the radius of the first 

dark ring in the image formed in the focal plane of an ordinary field glass. 

The diameter of the objective is 4 cm and its focal length 30 cm. White 

light has an effective wavelength of 5.6 X 10 -B cm, so that the angular 

5 6V 10 -5 
radius of this ring is 6 = 1.220 * = 1.71 X 10~ 5 rad. The 

linear radius is this angle multiplied by the focal length and there- 
fore amounts to 30 X 1.71 X 10 -5 = 0.000512 cm, or almost exactly 
0.005 mm. The central disk for this telescope is then 0.01 mm in diam- 
eter when the object is a point source such as a star. 

Extending Rayleigh's criterion for the resolution of diffraction patterns 
(Sec. 15.6) to the circular aperture, two patterns are said to be resolved 
when the central maximum of one falls on the first dark ring of the other. 
The resultant pattern in this condition is shown in Fig. \bK(b). The 
minimum angle of resolution for a telescope is therefore 

di = 1.220 -^ (15i) 

where D is the diameter of the circular aperture which limits the beam 
forming the primary image, or usually that of the objective. For the 
example chosen above, the angle calculated is just this limiting angle, 
so that the smallest angular separation of a double star which could be 
theoretically resolved by this telescope is 1.71 X 10~ 5 rad, or 3.52 seconds 
of arc. Since the minimum angle is inversely proportional to D, we see 
that the aperture necessary to resolve two sources 1 second apart is 
3.52 times as great as in the example, or that 

14 1 
Minimum angle of resolution in seconds 0i = — jr- (15j) 

D being the aperture of the objective in centimeters. For the larg- 
est refracting telescope in existence, that at the Yerkes Observatory, 
D = 40 in. and 0i = 0.14 sec. This may be compared with the minimum 
angle of resolution for the eye, the pupil of which has a diameter of about 
3 mm. We find 0i = 47 seconds of arc* Actually the eye of the aver- 
age person is not able to resolve objects less than about 1 minute apart, 
and the limit is therefore effectively determined by optical defects in 
the eye <5r by the structure of the retina. 

With a given objective in a telescope, the angular size of the image as 

* It might at first appear that the wavelength to be used in this calculation would 
be that in the vitreous humor of the eye. It is true that the dimensions of the diffrac- 
tion pattern are smaller on this account, but the separation of two images is also 
decreased in the same proportion by refraction of the rays as they enter Vhe eye. 



FRAUNHOFER DIFFRACTION BY A SINGLE OPENING 305 

seen by the eye is determined by the magnification of the eyepiece. 
However, increasing the size of the image by increasing the power of 
the eyepiece does not increase the amount of detail that can be seen, 
since it is impossible by magnification to bring out detail which is not 
originally present in the primary image. Each point in an object becomes 
a small circular diffraction pattern or disk in the image, so that if an 
eyepiece of very high power is used, the image appears blurred and no 
greater detail is seen. Thus diffraction by the objective is the one factor 
that limits the resolving power of a telescope. 

The diffraction pattern of a circular aperture, as well as the resolving 
power of a telescope, can be demonstrated by an experimental arrange- 
ment similar to that shown in Fig. 15/7. The point sources at S t and 
Sz consist of a sodium or mercury arc and a screen with several pinholes 
about 0.35 mm in diameter and spaced from 2 to 10 mm apart. These 
may be viewed with one of three small holes 1, 2, and 4 mm in diameter, 
mounted in front of the objective lens to show how an increasing aper- 
ture affects the resolution. Under these circumstances the intensity will 
not be sufficient to show anything but the central disks. In order to 
observe the subsidiary diffraction rings, the best source to use is the 
concentrated-arc lamp to be described in Sec. 21.2. 

The theoretical resolving power of a telescope will be realized only if 
the lenses are geometrically perfect and if the magnification is at least 
equal to the so-called "normal" magnification (Sec. 7.14). To prove 
the latter statement, we note that two diffraction disks which are on 
the limit of resolution in the focal plane of the objective must subtend 
at the eye an angle of at least 6[ = l.22X/d e in order to be resolved by 
the eye. Here d e represents the diameter of the eye pupil. Now accord- 
ing to Eq. \0k the magnification 

., 6' D 

6 d 

where D is the diameter of the entrance pupil (objective) and d that of 
the exit pupil. At the normal magnification, d is made equal to d e , so 
that the normal magnification becomes 

D _ 1.22\/d e = d[ 
d e 1.22X/D di 

Hence, if the diameter d of the exit pupil is made larger than d e , that 
of the eye pupil, we have 6' < 6[ and the images will cease to be resolved 
by the eye even though they are resolved in the focal plane of the objec- 
tive. In other words, any magnification that is less than the normal one 
corresponds to an exit pupil larger than d e , and will not afford the resolu- 
tion that the instrument could give. 



306 PHYSICAL OPTICS 

15.10. Brightness and Illuminance of Star Images. It was proved 
in Sec. 7.13 that regardless of the aperture of an instrument, for magni- 
fications up to the normal magnification the brightness of the image of 
an extended object remains constant and at most equal to that of the 
object. If the object is a point source this is no longer true, but instead 
the brightness increases rapidly for larger apertures. This is because 
all the light collected by the objective is concentrated in a diffraction 
pattern at its focal plane, and the area of this pattern varies inversely 
as the square of the diameter of the objective (Eq. 15i). Assuming 
normal magnification or greater, all light from the objective is admitted 
by the eye pupil, and the increase in brightness due to the telescope 
therefore equals the ratio of the area of the objective to that of the eye 
pupil. If the magnification is less than the normal, the eye constitutes 
the aperture stop and the exit pupil, and its image formed by the tele- 
scope is the entrance pupil. The ratio of their areas is the square of 
the magnification of the telescope, which then gives the factor by which 
the brightness is increased. The area of the retina illuminated remains 
constant, since it is determined by the diffraction pattern produced by 
the pupil of the eye. 

The illuminance of the image of a point source may be calculated by 
multiplying the illuminance of the objective by the ratio of its area to 
that of the central disk of the diffraction pattern it produces, because 
most of the light entering the objective goes into this disk. Thus the 
illuminance will be proportional to the area of the objective. It is 
chiefly for this reason that attempts are constantly being made to 
increase the diameter of telescope objectives. The 200-in. mirror of 
the Mt. Palomar telescope permits the photography of much fainter 
objects than has heretofore been possible. 

15.11. Resolving Power of a Microscope. In this case the same 
principles are applicable. The conditions are, however, different from 
those for a telescope, in which we were chiefly interested in the smallest 
permissible angular separation of two objects at a large, and usually 
unknown, distance. With a microscope the object is very close to the 
objective, and the latter subtends a large angle 2ii at the object plane, 
as shown in Fig. 15L. Here we wish primarily to know the smallest 
distance between two points and 0' in the object which will produce 
images / and V that are just resolved. Each image consists of a disk 
and a system of rings, as explained above, and the angular separation of 
two disks when they are on the limit of resolution is a =*= d x = 1.22X/D. 
When this condition holds, the wave from 0' diffracted to / has zero 
intensity (first dark ring), and the extreme rays O'BI and O'AI differ in 
path by 1.22X. From the insert in Fig. 15L, we see that O'B is longer 
than OB or OA by s sin i, and 0' A shorter by the same amount. The 
path difference of the extreme rays from 0' is thus 2s sin i, and upon 



FRAUNHOFER DIFFRACTION BY A SINGLE OPENING 

equating this to 1.22X, we obtain 

1.22X 



s = 



2 sin i 



307 



(15fc) 



In this derivation, we have assumed that the points and 0' were self- 
luminous objects, such that the light given out by each has no constant 
phase relative to that from the other. Actually the objects used in 
microscopes are not self-luminous but are illuminated with light from a 




Fig. 15/y. Resolving power of a microscope. 

condenser. In this case it is impossible to have the light scattered by 
two points on the object entirely independent in phase. This greatly 
complicates the problem, since the resolving power is found to depend 
somewhat on the mode of illumination of the object. Abbe investigated 
this problem in detail and concluded that a good working rule for calcu- 
lating the resolving power was given by Eq. 15fc, omitting the factor 
1.22. In microscopes of high magnifying power, the space between the 
object and the objective is filled with an oil. Beside decreasing the 
amount of light lost by reflection at the first lens surface, this increases 
the resolving power, because when refraction of the rays emerging from 
the cover glass is eliminated, the objective receives a wider cone of light 
from the condenser. Equation 15k must then be further modified by 
substitution of 2ns sin i for the optical path difference, where n is the 
refractive index of the oil. The result of making these two changes is 



s = 



2n sin i 



(151) 



The product n sin i is characteristic of a particular objective, and was 
called by Abbe the "numerical aperture." In practice the largest value 
of the numerical aperture obtainable is about 1.6. With white light of 



308 PHYSICAL OPTICS 

effective wavelength 5.6 X 10 -B cm, Eq. 15Z gives s = 1.8 X 10 -5 cm. 
The use of ultraviolet light, with its smaller value of X, has recently been 
applied to still further increase the resolving power. This necessitates 
the use of photography in examining the image. 

One of the most remarkable steps in the improvement of microscopic 
resolution has been the recent development of the electron microscope. 
As will be explained in Sec. 30.4, electrons behave like waves whose 
wavelength depends on the voltage through which they have been accel- 
erated. For voltages between 100 and 10,000 volts, X varies from 
1.22 X 10~ 8 to 1.22 X 10~ 9 cm, i.e., it lies in the region of a fraction of 
an angstrom unit. This is more than a thousand times smaller than for 
visible light. It is possible by means of electric and magnetic fields to 
focus the electrons emitted from, or transmitted through, the various 
parts of an object, and in this way details not very much larger than the 
wavelength of the electrons can be photographed. The numerical aper- 
ture of electron microscopes is still much smaller than that of optical 
instruments, but further developments in this large and growing field of 
electron optics are to be anticipated.* 

15.12. Phase Contrast. The eye readily detects differences in ampli- 
tude by intensity changes, but it is not able to see changes in phase 
directly. Thus, as long as the objects on a microscope slide are opaque 
or absorbing, they appear in the image. If they are transparent, how- 
ever, and differ only slightly from their surroundings in refractive index 
or in thickness, they will ordinarily not be visible. It is nevertheless 
possible to convert the phase changes produced by such objects into 
amplitude changes in the final image. The so-called phase-contrast 
microscope, devised in 1935 by Zernike,f functions in this way. 

To illustrate the basic principle involved here, let us consider how the 
alternately positive and negative phases in the successive maxima of the 
single-slit pattern (Fig. 15D) might be rendered visible. Suppose that 
one were to superimpose on the pattern as it appears on the screen a uni- 
form plane wave that is coherent with the waves forming the pattern, and 
therefore capable of interfering with them. If this additional wave 
were to be in phase with the light of the central maximum, we would 
produce constructive interference and increased intensity of this, and 
also of the second, fourth, etc., subsidiary maxima. The odd-numbered 
secondary maxima, however, would be out of phase with it, and the 
interference would be destructive. Zernike has shown how this effect 
may be produced experimentally by placing over a fairly wide slit a 

* See, for example, V. K. Zworykin, G. A. Morton, and others, "Electron Optics 
and the Electron Microscope," John Wiley & Sons, Inc., New York, 1945. 

t F. Zernike (1888- ). Professor of physics at the University of Groningen, 
Holland. In 1953 he was awarded the Nobel prize for his discovery of the phase- 
contrast principle. 



FRAUNIIOFER DIFFRACTION BY A SINGLE OPENING 309 

much narrower one with semitransparent jaws. The central maximum 
due to the latter is made broad enough to spread over the whole pattern 
caused by the wider slit, and its intensity may be adjusted so that it 
almost completely eliminates the alternate secondary maxima. The sup- 
pression of these, and the enhancement of the intermediate ones are direct 
evidence of the phase differences which were present in the original single- 
slit pattern, and which were otherwise quite unrecognizable to the eye. 

The way in which the principle of phase contrast is employed in the 
microscope is rather involved, and its explanation would require a 
lengthier discussion than is justified here.* It must suffice to say that 
the interference occurs between the direct light which passes unaffected 
through the uniform parts of the slide and the light which is diffracted 
by its irregular portions. The former consists of parallel beams, and is 
brought to a focus in the secondary focal plane of the objective, while the 
latter is focused in the plane of the image conjugate to the object. By 
placing a quarter-wave retarding plate called the "phase plate" in the 
secondary focal plane, the phase of the direct light, which is spread 
uniformly over the image plane, is altered in the proper way to produce 
an amplitude modulation in this plane which is proportional to the phase 
modulation caused by the object. In this way details of transparent 
biological specimens become visible as increases or decreases of intensity. 

PROBLEMS 

1. It is a general rule for any spectroscope in which the resolving power is limited 
by diffraction that 

Chromatic resolving power = angular dispersion X width of emerging beam 

Using Eq. 23c for the dispersion of a prism at minimum deviation, show that Eq. 15h 
agrees with this rule. 

2. Plane waves of wavelength 5461 A are incident normally on a slit which has a 
lens of focal length 40.0 cm behind it. If the width of the slit is 0.450 mm, find the 
distance from the principal maximum to the first minimum in the diffraction pattern 
formed in the focal plane of the lens. Ans. 0.485 mm. 

3. A slit 0.20 mm wide is illuminated perpendicularly by an intense parallel beam 
of white light. Two meters behind it a small spectroscope is used to explore the 
spectrum of the diffracted light. Predict what would be seen when the slit of the 
spectroscope is displaced in a direction perpendicular to the diffracting slit by a 
distance of 1 cm from the axis. 

4. When diffraction of parallel light by a slit is observed without a lens as in Fig. 
152?, the pattern will be essentially a Fraunhofer one when the distance of observation 
is at least equal to the square of the slit width divided by the wavelength. According 
to the description of the conditions used in photographing the pattern of Fig. 15B, 
how far from the diffracting slit would the plate have to be placed to photograph such 
a pattern without using the lens L»? Ans. 1.86 cm. 

* A complete discussion of the subject will be found in A. H. Bennett, H. Jupnik, 
H. Osterberg, and O. W. Richards, "Phase Microscopy," John Wiley & Sons, Inc., 
ftew York, 1951. 



S10 PHYSICAL OPTICS 

6. Carry through the derivation of Eq. 15d by the method of complex amplitudes 
as suggested in the footnote, page 291. 

6. Make an accurate plot of the intensity in the Fraunhofer diffraction pattern 
of a slit in the region of the first subsidiary maximum (/3 = * to 2*-). From your 
graph, verify the figures given in Table 15-1 for the position and intensity of this 
maximum. Ans. At = 1.430*-, a maximum, of intensity 4.72%. 

7. Calculate the approximate intensity of the first weak maximum that appears 
along the diagonal 0/y = l/b in the Fraunhofer diffraction pattern of a rectangular 
aperture. 

8. Considering the criterion for the resolution of two diffraction patterns of unequal 
intensity to be that the drop in intensity between the maxima shall be 20 per cent 
of the weaker one, find the angular separation required when the intensities are in 
the ratio 5:1. Express the result in terms of U the angle required when the intensi- 
ties are equal. This problem may best be solved graphically, using two plots that may 
be superimposed with a variable displacement. Ans. 1.1301. 

9. From the data given in Table 23-1 for barium flint glass, calculate the chromatic 
resolving power of a 60° equiangular prism of this material, if the width of the sides 
is 6 cm. Make the calculation for the wavelength of the sodium D lines, and for 
that of the calcium H line. 

10. It is desired to resolve a double spectrum line in the ultraviolet, the wave- 
lengths of the components being known to be 3130.326 and 3130.409 A. A spectro- 
graph containing a crystalline quartz prism with a 10-cm base is available. Such a 
prism is always made so that the refractive index no of Table 26-1 is the effective one. 
Find whether it is theoretically possible for this spectrograph to resolve the doublet. 

Ans. Not resolved. 

11. Carry through the differentiation of Eq. 15c and prove that tan = is the 
condition for maxima. 

^/Find the diameter of the first bright ring (secondary maximum) in the focal 
plane of the 36-in. refractor of the Lick Observatory. The focal length is 56 ft, and 
the effective wavelength of white light may be taken as 5500 A. Ans. 0.0336 mm. 

13. What is the maximum permissible width of the slit source according to the 
criterion stated at the end of Sec. 15.5 under the following circumstances: source to 
diffracting slit 50 cm; width of diffracting slit 0.5 mm; wavelength 6000 A? 

14. The pupil of the eye has an average diameter of 2.5 mm in daylight. At what 
distance would two small orange-colored objects (X = 6000 A) 40 cm apart be barely 
resolved by the naked eye, assuming the resolution to be limited by diffraction only? 

^_^ Ans. 1.37 km. 

(45^ In the projection of a beam of underwater sound for submarine detection, a circu- 
lar diaphragm 50 cm in diameter is made to oscillate at a frequency of 30,000 cycles, sit. 
At some distance from such a source, the intensity distribution will be the Fraun- 
hofer pattern for a circular hole of diameter equal to that of the diaphragm. Find 
the angle between the normal and the first minimum for the given frequency, and 
also for the audible frequency of 1200 per sec. Assume the velocity of sound to be 
1435 m/sec. 

16. Find the numerical aperture of the microscope objective that would be required 
to resolve the lines on a diffraction grating ruled with 14,438 lines to the inch using 
sodium light. If the objective were to be an immersion one using an oil having 
no = 1.50, what would be the required angle of the cone of light to fill the objective? 

Ans. 0.167. 12°49'. 

17. Calculate the minimum angle of resolution in seconds of arc for a small gal- 
vanometer telescope, the objective of which is 12 mm in diameter. What magnifica- 
tion would be required in order for this resolution to be achieved? 



CHAPTER 16 

THE DOUBLE SLIT 



The interference of light from two narrow slits close together was first 
demonstrated by Young, and it has already been discussed in Sec. 13.2 
as a simple example of the interference of two beams of light. In our 
discussion of the experiment, the slits were assumed to have widths not 
much greater than a wavelength of light, so that the central maximum 
in the diffraction pattern from each slit separately was wide enough to 
occupy a large angle behind the screen (Figs. ISA and 132?) . It is 
important to understand the modifications of the interference pattern 
which occur when the width of the individual slits is made greater, until it 
becomes comparable with the distance between them. This corresponds 
more nearly to the actual conditions under which the experiment is usually 
performed. In this chapter we shall discuss the Fraunhofer diffraction 
by a double slit, and some of its applications. 

16.1. Qualitative Aspects of the Pattern. In Fig. 16 A (6) and (c) 
photographs are shown of the patterns obtained from two different double 
slits in which the widths of the individual slits were equal in each pair, 
but where the two pairs were different. Referring to Fig. 16B, which 
shows the experimental arrangement for photographing these patterns, 
the slit width b of each slit was greater for Fig. 16 A (c) than for Fig. 
lQA(b), but the distance between centers d = b -f- c, or the separation 
of the slits, was the same in the two cases. In the central part of Fig. 
16.A (6) are seen a number of interference maxima of approximately uni- 
form intensity, resembling the interference fringes described in Chap. 13 
and shown in Fig. 13D. The intensities of these maxima are not actually 
constant, however, but fall off slowly to zero on either side and then 
reappear with low intensity two or three times before becoming too 
faint to observe without difficulty. The same changes are seen to occur 
much more rapidly in Fig. 16 A (c), which was taken with the slit widths 
b somewhat larger. 

16.2. Derivation of the Equation for the Intensity. Following the 
same procedure as that used for the single slit in Sec. 15.2, it is merely 
necessary to change the limits of integration in Eq. 156 to include the 

311 



312 



PHYSICAL OPTICS 




Fig. 16^4. Diffraction patterns from (a) a single narrow slit, (b) two narrow slits, (c) 
two wider slits, (d) one wider slit. 



two portions of the wave front transmitted by the double slit.* Thus if 
we have, as in Fig. 165, two equal slits of width 6, separated by an opaque 
space of width c, the origin may be chosen at the center of c, and the 
integration extended from s = (d/2) - (6/2) to s = (d/2) + (6/2). 
This gives 



y = 



2a 



xk sin 



[ sin 6 



5 k(d + b) sin 6 ) - 



) - sin (5 



5 k(d — 6) sin 6 



[sin (at — kx)] 



The quantity in brackets is of the form sin (A + B) — sin (A — B), 
and when it is expanded, we obtain 



where, as before, 



26a sin /3 , . , . 

y = rr- cos 7 sin (wl — kx) 

x p 



j8 = 7T kb sin 6 = - 6 sin 6 

2t A 



(16a) 



* The result of this derivation is obviously a special case of the general formula 
for N slits, which will be obtained by the method of complex amplitudes in the follow- 
ing chapter. 



THE DOUBLE SLIT 



313 



and where 



7 = jr k(b + c) sin 6 = r- d sin 6 



(166) 



The intensity is proportional to the square of the amplitude of Eq. 16a, 
so that, replacing ba/x by A as before, we have 



I = 4A, 



sin 2 



cos' 7 



(16c) 



The factor (sin 2 /3)//3 2 in this equation is just that derived for the single 





Source 



Double slit Screen 

Fig. 16B. Apparatus for observing Fraunhofer diffraction from a double slit. Drawn 
for the case 26 = c, that is, d = 36. 

slit of width 6 in the previous chapter (Eq. 15d). The second factor 
cos 2 y is characteristic of the interference pattern produced by two beams 
of equal intensity and phase differ- 
ence 5, as shown in Eq. 136 of Sec. 
13.3. There the resultant intensity 
was found to be proportional to 
cos 2 (5/2), so that the expressions 
correspond if we put y = 8/2. The 
resultant intensity will be zero when 
either of the two factors is zero. 
For the first factor this will occur 
when = 7r, 2tt, St, . . . , and for 
the second factor when y = ir/2, 
3tt/2, 5V/2, .... That the two 
variables and y are not independ- 
ent will be seen from Fig. 16C. The 
difference in path from the two edges 
of a given slit to the screen is, as indicated, 6 sin 6. The corresponding 
phase difference is, by Eq. 11/, (27r/X) b sin 6, which equals 2/3. The 
path difference from any two corresponding points in the two slits is, as 




Fig. 16C. Path differences of parallel rays 
leaving a double slit. 



314 PHYSICAL OPTICS 

is illustrated for the two points at the lower edges of the slits, d sin 6, 
and the phase difference 8 = (2w/\)d sin 6 = 2?. Therefore, in terms 
of the dimensions of the slits, 

16.3. Comparison of the Single-slit and Double-slit Patterns. It is 

instructive to compare the double-slit pattern with that given by a single 
slit of width equal to that of either of the two slits. This amounts to 
comparing the effect obtained with the two slits in the arrangement 
shown in Fig. 16B with that obtained when one of the slits is entirely 
blocked off with an opaque screen. If this is done, the corresponding 
single-slit diffraction patterns are observed, and they are related to the 
double-slit patterns as shown in Fig. 164 (a) and (d). It will be seen 
that the intensities of the interference fringes in the double-slit pattern 
correspond to the intensity of the single-slit pattern at any point. If 
one or other of the two slits is covered, we obtain exactly the same single- 
slit pattern in the same position, while if both slits are uncovered the 
pattern, instead of being a single-slit one with twice the intensity, breaks 
up into the narrow maxima and minima called interference fringes. The 
intensity at the maximum of these fringes is four times the intensity of 
either single-slit pattern at that point, while it is zero at the minima 
(see Sec. 13.4). 

16.4. Distinction between Interference and Diffraction. One is quite 
justified in explaining the above results by saying that the light from 
the two slits undergoes interference to produce fringes of the type obtained 
with two beams, but that the intensities of these fringes are limited by 
the amount of light arriving at the given point on the screen by virtue 
of the diffraction occurring at each slit. The relative intensities in the 
resultant pattern as given by Eq. 16c are just those obtained by multi- 
plying the intensity function for the interference pattern from two 
infinitely narrow slits of separation d (Eq. 136) by the intensity function 
for diffraction from a single slit of width b (Eq. I5d). Thus, the result 
may be regarded as due to the joint action of interference between the 
rays coming from corresponding points in the two slits and of diffraction, 
which determines the amount of light emerging from either slit at a given 
angle. But diffraction is merely the result of the interference of all the 
secondary wavelets originating from the different elements of the wave 
front. Hence it is proper to say that the whole pattern is an interfer- 
ence pattern. It is just as correct to refer to it as a diffraction pattern, 
since, as we saw from the derivation of the intensity function in Sec. 
16.2, it is obtained by direct summing the effects of all of the elements 
of the exposed part of the wave front. However, if we reserve the term 



THE DOUBLE SLIT 315 

interference for those cases in which a modification of amplitude is pro- 
duced by the superposition of a finite (usually small) number of beams, 
and diffraction for those in which the amplitude is determined by an 
integration over the infinitesimal elements of the wave front, the double- 
slit pattern can be said to be due to a combination of interference and 
diffraction. Interference of the beams from the two slits produces the 
narrow maxima and minima given by the cos 2 7 factor, and diffraction, 
represented by (sin 2 /3)//3 2 , modulates the intensities of these interference 
fringes. The student should not be misled by this statement into think- 
ing that diffraction is anything other than a rather complicated case of 
interference. 

16.5. Positions of the Maxima and Minima. Missing Orders. As 
shown in Sec. 16.2, the intensity will be zero wherever 7 = ir/2, 37r/2, 
5tt/2, . . . and also when = ir, 2tt, 3ir, .... The first of these two 
sets are the minima for the interference pattern, and since by definition 
7 = (ir/\)d sin 0, they occur at angles such that 

d sin 8 — «»*9*» "«■»•• ■ — ( fl* + slX minima (16e) 

m being any whole number starting with zero. The second series of min- 
ima are those for the diffraction pattern, and these, since /3 = (w/X)a sin 0, 
occur where 

6 sin = X, 2X, 3X, . . . = p\ minima (16/) 

the smallest value of p being 1. The exact positions of the maxima are 
not given by any simple relation, but their approximate positions may 
be found by neglecting the variation of the factor (sin 2 /3)//3 2 , a justified 
assumption only when the slits are very narrow and when the maxima 
near the center of the pattern are considered [Fig. 16.4(6)]. The posi- 
tions of the maxima will then be determined solely by the cos 2 7 factor, 
which has maxima for 7 = 0, tt, 2ir, . . . , i.e., for 

d sin = 0, X, 2X, 3X, . . . = raX maxima (16gr) 

The whole number m represents physically the number of wavelengths 
in the path difference from corresponding points in the two slits (see 
Fig. 16C) and represents the order of interference. 

Figure 16D(a) is a plot of the factor cos 2 7, and here the values of the 
order, of half the phase difference 7 = 5/2, and of the path difference 
are indicated for the various maxima. These are all of equal intensity 
and equidistant on a scale of d sin 0, or practically on a scale of 0, since 
when is small sin &i and the maxima occur at angles = 0, \/d, 
2\/d, .... With a finite slit width b the variation of the factor 
(sin 2 /3)//3 2 must be taken into account. This factor alone gives just the 



310 



PHYSICAL OPTICS 



single-slit pattern discussed in the last chapter, and is plotted in Fig. 
16-0(6). The complete double-slit pattern as given by Eq. 16c is the 
product of these two factors, and therefore is obtained by multiplying 
the ordinates of curve (a) by those of curve (6) and the constant 4A 2 . 
This pattern is shown in Fig. 16D(c). The result will depend on the rela- 
tive scale of the abscissas and 7, which in the figure are chosen so that for 
a given abscissa 7 = 3/3. But the relation between and 7 for a given 




m = -8 -7 


-6 


-5 


-4 


-3 


-2 


-1 





1 


2 


3 


4 


5 


6 


7 


8 


9 


P = 


-2 






-1 













1 






2 






3 



Fig. 16Z). Intensity curves for a double slit where d = 36. 

angle 6 is determined, according to Eq. 10>d, by the ratio of the slit 
width to the slit separation. Hence if d = 36, the two curves (a) 
and (6) are plotted to the same scale of 0. For the particular case of 
two slits of width b separated by an opaque space of width c = 26, the 
curve (c), which is the product of (a) and (6), then gives the resultant 
pattern. The positions of the maxima in this curve are slightly different 
from those in curve (a) for all except the central maximum (m = 0), 
because when the ordinates near one of the maxima of curve (a) are 
multiplied by a factor which is decreasing or increasing, the ordinates 
on one side of the maximum are changed by a different amount from 
those of the other, and this displaces the resultant maximum slightly 
in the direction in which the factor is increasing. Hence the positions 



THE DOUBLE SLIT 317 

of the maxima in curve (c) are not exactly those given by Eq. 16<?, but 
in most cases will be very close to them. 

Let us now return to the explanation of the differences in the two pat- 
terns (6) and (c) of Fig. 16A, taken with the same slit separation d but 
different slit widths 6. Pattern (c) was taken for the case d = 36, and 
is seen to agree with the description given above. For pattern (6), the 
slit separation d was the same, giving the same spacing for the interfer- 
ence fringes, but the slit width 6 was smaller, such that d = 66. In 
Fig. 13Z>, d = 146. This greatly increases the scale for the single-slit 
pattern relative to the interference pattern, so that many interference 
maxima now fall within the central maximum of the diffraction pattern. 
Hence the effect of decreasing 6, keeping d unchanged, is merely to 
broaden out the single-slit pattern, which acts as an envelope of the 
interference pattern as indicated by the dotted curve of Fig. 16Z>(c). 

If the slit-width 6 is kept constant and the separation of the slits d 
is varied, the scale of the interference pattern varies, but that of the 
diffraction pattern remains the same. A series of photographs taken to 
illustrate this is shown in Fig. 1QE. For each pattern three different 
exposures are shown, to bring out the details of the faint and the strong 
parts of the pattern. The maxima of the curves are labeled by the 
order m, and underneath the upper one is a given scale of angular posi- 
tions 6. A study of these figures shows that certain orders are missing, 
or at least reduced to two maxima of very low intensity. These so-called 
missing orders occur where the condition for a maximum of the inter- 
ference, Eq. 16<7, and for a minimum of the diffraction, Eq. 16/, are both 
fulfilled for the same value of 0, that is for 

d sin 6 = m\ 
6 sin 6 = p\ 

so that t = ~ ( 16 *) 

Since m and p are both integers, d/b must be in the ratio of two integers 
if we are to have missing orders. This ratio determines the orders which 
are missing, in such a way that when d/b = 2, orders 2, 4, 6, . . . are 
missing; when d/b = 3, orders 3, 6, 9, . . . are missing; etc. When 
d/b = 1, the two slits exactly join, and all orders should be missing. 
However, the two faint maxima into which each order is split can then 
be shown to correspond exactly to the subsidiary maxima of a single-slit 
pattern of width 26. 

Our physical picture of the cause of missing orders is as follows: Con- 
sidering for example the missing order m = +3 in Fig. 16Z)(c), this point 
on the screen is just three wavelengths farther from the center of one 
slit than from the center of the other. Hence we might expect the waves 



318 



PHYSICAL OPTICS 



II 



Zb = d 




46. d 



■scaa i iTS'n , fl 




567 91O11 



< 1 



56 = d 


1 
1 
1 
/ 


ill II A 2 

If 1 ' 
\ 

III is 




•ACVH^J 


1) II (1 UUi/V .«aa.' | '.».!.\ 




789l0ii l3 14l5i6,7 
Fig. 162?. Photographs and intensity curves for double-slit diffraction patterns. 



THE DOUBLE SLIT 



319 



from the two slits to arrive in phase and to produce a maximum. How- 
ever, this point is at the same time one wavelength farther from the 
edge of one slit than from the other edge of that slit. Addition of the 
secondary wavelets from one slit gives zero intensity under these condi- 
tions. The same holds true for either slit, so that, although we may add 
the contributions from the two slits, both contributions are zero and 
must therefore give zero resultant. 




' A l ' 



^ >£ 



abed e f g h i 
6=0 \ n 1% 2jr5%37r7? 2 4/r 

/3=o \ V% V% 3 V% * 



ky^ ©^ 



(c) 

. 4 



U) 



& \ 



"o- 



(61 



P 



>*r 




*fcj 



•B 



(e) 



17 



If) (g) (A) (i) 

Fig. 16F. Illustrating how the intensity curve for a double slit is obtained by the 
graphical addition of amplitudes. 

16.6. Vibration Curve. The same method as that applied in Sec. 15.4 
for finding the resultant amplitude graphically in the case of the single 
slit is applicable to the present problem. For illustration we take a 
double slit in which the width of each slit equals that of the opaque 
space between the two, so that d = 26. A photograph of this pattern 
appears in Fig. 16E 1 at the top. A vector diagram of the amplitude 
contributions from one slit gives the arc of a circle, as before, the differ- 
ence between the slopes of the tangents to the arc at the two ends being 
the phase difference 2/3 between the contributions from the two edges 
of the slit. Such an arc must now be drawn for each of the two slits, 
and the two arcs must be related in such a way that the phases (slopes 
of the tangents) differ for corresponding points on the two slits by 2y, or 5. 
In the present case, since d = 26, we must have y = 2/3 or 8 = 4/3. Thus 
in Fig. 16F(6) showing the vibration curve for /3 = v/8, both arcs sub- 
tend an angle of 7r/4 (= 2/3), the phase difference for the two edges of 
each slit, and the arcs are separated by ?r/4 so that corresponding points 



320 PHYSICAL OPTICS 

on the two arcs differ by ir/2(= 5). Now the resultant contributions 
from the two slits are represented in amplitude and phase by the chords 
of these two arcs, that is by A a and A 2 . Diagrams (a) to (i) give the 
construction for the points similarly labeled on the intensity curve. The 
intensity, it will be remembered, is found as the square of the resultant 
amplitude A, which is the vector sum of Ai and Ai. 

In the example chosen, the slits are relatively wide compared with 
their separation,, and as the phase difference increases the curvature of 
the individual arcs of the vibration curve increases rapidly, so that the 
vectors Ai and A 2 decrease rapidly in length. For narrower slits we 
obtain a greater number of interference fringes within the central diffrac- 
tion maximum, because the lengths of the arcs are smaller relative to 
the radius of curvature of the circle. A\ and A 2 then decrease in length 
more slowly with increasing /3, and the intensities of the maxima do not 
fall off so rapidly. In the limit where the slit width a approaches zero, 
A i and A 2 remain constant, and the variation of the resultant intensity 
is merely due to the change in phase angle between them. 

16.7. Effect of Finite Width of Source Slit. A simplification which 
was made in the above treatment, and which never holds exactly in 
practice, is the assumption that the source slit (S' of Fig. 16B) is of 
negligible width. This is necessary in order that the lens shall furnish 
a single train of plane waves falling on the double slit. Otherwise there 
will be different sets of waves approaching at slightly different angles, 
these originating from different points in the source slit. They will 
produce sets of fringes which are slightly shifted with respect to each 
other, as illustrated in Fig. 16G(a). In the figure the interference maxima 
are for simplicity drawn with uniform intensity, neglecting the effects 
of diffraction. Let P and P' be two narrow lines acting as sources. 
These may be two narrow slits, or, better, two lamp filaments, since we 
assume no coherence between them. If the positions of the central 
maxima of the interference patterns produced by these are Q and Q', 
the fringe displacement QQ' will subtend the same angle a at the double 
slit as do the source slits. If this angle is a small fraction of the angular 
separation X/d of the successive fringes in either pattern, the resultant 
intensity distribution will still resemble a true cos 2 y curve, although the 
intensity will not fall to zero at the minima. The relative positions 
of the two patterns, and the sum of the two, in this state are illustrated 
in Fig. 1QG, curves (6). Curves (c) and (d) show the effect of increasing 
the separation PP'. For (d) the fringes are completely out of step, and 
the resultant intensity shows no fluctuations whatever. At a point such 
as Q the maximum of one pattern then coincides with the next minimum 
of the other, so that the path difference P'AQ — PAQ = X/2. In other 
words, P' is just a half wavelength farther from A than is P. If the 



THE DOUBLE SLIT 



321 



intensity of one set of fringes is given by 4 A 2 cos 2 (5/2) or 2A 2 (1 + cos 5), 
that of the other is 

2A 2 [1 + cos (5 + t)] = 2/1 2 (1 - cos 5) 

The sum is therefore constant and equal to 4A 2 , so that the fringes 
entirely disappear. The condition for this disappearance of fringes is 
a = X/2d. If PP' is still further increased, the fringes will reappear, 




Components 

A. A A A , YMYXYX YXmXXY 

Fig. 16G. Effect of a double source and of a wide source on the double-slit interference 
fringes. 

becoming sharp again when a equals the fringe distance \/d, disappear 
when the fringes are again out of register, etc. In general, the condition 
for disappearance is 



\_ 3X 5X 

2d' 2d' 2d' 



DISAPPEARANCE OF FRINGES WITH 
DOUBLE SOURCE 



(16») 



where a is the angle subtended by the two sources at the double slit. 

Next let us consider the effect when the source, instead of consisting 
of two separate sources, consists of a uniformly bright strip of width 
PP'. Each line element of this strip will produce its own set of inter- 
ference fringes, and the resultant pattern will be the sum of a large 
number of these, displaced by infinitesimal amounts with respect to each 
other. Figure 10G(e) illustrates this for a = \/2d, that is, for a slit of width 
such that the extreme points acting alone would give complete disappear- 
ance of fringes as in (d). The resultant curve now shows strong fluctua- 
tions, and the slit must be still further widened to make the intensity 



322 PHYSICAL OPTICS 

uniform. The first complete disappearance will come when the range 
covered by the component fringes extends over a whole fringe width, 
instead of one-half, as above. This case is shown in Fig. lGG(f), for a 
slit of width subtending an angle a = \/d. Widening the slit still further 
will cause the fringes to reappear, although they never become perfectly 
distinct again, with zero intensity between fringes. At a = 2X/d they 
again disappear completely, and the general condition is 

_ X 2X 3X DISAPPEARANCE OF FRINGES WITH SLIT 

01 ~ d! d ' ~d ' * • ' source (16;) 

It is of practical importance, in observing double-slit fringes experi- 
mentally, to know how wide the source slit may be made in order to 
obtain intense fringes without seriously impairing the definition of the 
fringes. The exact value will depend on our criterion for clear fringes, 
but a good working rule is to permit a maximum discordance of the 
fringes of about one-quarter of that for the first disappearance. If /' 
is the focal length of the first lens, this corresponds to a maximum per- 
missible width of the source slit 

PP ' =f ' a= M < 16fc > 

16.8. Michelson's Stellar Interferometer. As was shown in Sec. 15.9, 
the smallest angular separation that two point sources may have in order 
to produce images which are recognizable as separate, in the focal plane 
of a telescope, is a = d x = 1.22X/D. In this equation (Eq. I5i) D is 
the diameter of the objective of the telescope. Suppose that the objec- 
tive is covered by a screen pierced with two parallel slits of separa- 
tion almost equal to the diameter of the objective. A separation of 
d = D/1.22 would be a convenient value. If the telescope is now pointed 
at a double star and the slits are turned so as to be perpendicular to the 
line joining the two stars, interference fringes due to the double slit will in 
general be observed. However, according to Eq. 16i, if the angular 
separation of the two stars happens to be a = X/2d, the condition for the 
first disappearance, no fringes will be seen. Those from one star com- 
pletely mask those from the other. Hence one could infer from the non- 
appearance of the fringes that the star was double with an angular 
separation \/2d or some multiple of this. (The multiples could be ruled 
out by direct observation without the double slit.) But this separation 
is only half as great as the minimum angle of resolution of the whole 
objective 1.22X/D which in this case equals \/d. In this connection it is 
instructive to compare, as in Fig. 16/7, the dimensions of the diffraction 
pattern due to a rectangular aperture of width b with the interference 
pattern due to two narrow slits whose separation d is equal to 6- The 



THE DOUBLE SLIT 



323 



central maximum is only half as wide in the second case. Hence it is 
sometimes said that the resolving power of a telescope may be increased 
twofold by placing a double slit over the objective. This statement needs 
two important qualifications, however. In the first place the stars are not 
"resolved" in the sense of producing separate images, but their existence is 
merely inferred from the behavior of the fringes. In the second place, a 
partial blurring of the fringes, without complete disappearance, can be ob- 
served for separations much less than \/2d, showing the existence of two 
stars, and from this point of view the minimum resolvable separation is 



( 



T 
i 




(a) 



-X 



\/2d 




JL 



%■ 




ib) 



Fig. 16H. Fraunhofer pattern from (a) a rectangular aperture and (b) double slit 
of separation equal to the width of the aperture in (c). In (6) are shown the four 
auxiliary mirrors used in the actual stellar interferometer. 

considerably smaller than that indicated by the above statement. In 
practice it is about one-tenth of this. 

The actual measurement of the separation of a given close double star 
is made by having the slit separation d adjustable. The separation is 
increased until the fringes first disappear; then, by measuring d, the 
angular separation is obtained as a. = \/2d. The effective wavelength 
X of the starlight must, of course, be also estimated or measured. Sep- 
arations of double stars are not often determined by this method, because 
the advantage over the direct method is not very great, and observations 
on the Doppler effect (Sec. 11.6) afford an even more sensitive means of 
detection and measurement. On the other hand, the method of double- 
slit interference was, until recently,* the only way of measuring the 
diameter of the disk of a single star, and in 1920 this method was success- 
fully applied by Michelson for this purpose. 

From the discussion of the preceding section, it will be seen that if a 
source such as a star disk subtends a finite angle, disappearance of the 
fringes would be expected from this cause when the separation of the 
double slit on a telescope is made great enough. Michelson first demon- 

*See R. H. Brown and R. Q. Twiss, Nature, 178, 1447, 1956. 



324 PHYSICAL OPTICS 

strated the practicability of this method by measuring the diameters of 
Jupiter's moons, which subtend an angle of about one second. The values 
of d for the first disappearance are only a few centimeters in this case, 
and the measurement could be made by a double slit of variable separa- 
tion over the objective of a telescope. Owing to the fact that the source 
is a circular disk instead of a rectangle, a correction must be applied to 
the equation a = \/d for a slit source. This correction may be found 
by the same method that is used in finding the resolving power of a 
circular aperture, and gives the same factor. It is found that a = 1.22X/d 
gives the first disappearance for a disk source. Estimating the angular 
diameters of the nearer fixed stars of known distance by assuming they 
are the same size as the sun, one obtains angles less than 0.01 second. 
The separations of the double slit required to detect a disk of this size 
are from 20 to 40 ft. Clearly no telescope in existence could be used 
in the way described above for the measurement of star diameters. 
Another drawback would be that the fringes would be so fine that it 
would be difficult to separate them. 

Since the blurring of the fringes is the result of variations of the phase 
difference between the light arriving at the two slits from various points 
on the source, Michelson realized that it was possible to magnify this 
phase difference without increasing d. This was done by receiving 
the light from a star on two plane mirrors M and M' [Fig. lGH(b)] and 
reflecting it into the slits by these and two other mirrors. Then a 
variation of a in the angle of the incoming rays will cause a difference of 
path to the two slits of La, where L is the distance MM' between the outer 
mirrors. The fringes will now disappear when this difference equals 
1.22X, and so the sensitivity is magnified in the ratio L/d. In the actual 
measurements, M and M' were two 6-in. mirrors mounted on a girder in 
front of the 100-in. Mt. Wilson reflector so that they could be moved 
apart symmetrically. In the case of the star Arcturus, for example, the 
first disappearance of fringes occurred at L = 24 ft, indicating an angular 
diameter a = 1.22X/L of only 0.02 second. From the known distance of 
Arcturus, one then finds that its actual diameter is 27 times that of the 
sun.* 

16.9. Wide-angle Interference. Nothing has thus far been said as to 
whether there is any limit to the angle between the two interfering beams 
as they leave the source. Consider, for example, the double-slit arrange- 
ment shown in Fig. 167(a). The source S could be a narrow slit, but to 
ensure that there is no coherence between the light leaving various points 
on it, we shall assume that it is a self-luminous object. It is found 
experimentally that the angle 4> may be made fairly large without spoil- 

* Details of these measurements will be found in A. A. Michelson, "Studies in 
Optics," chap. 11, University of Chicago Press, Chicago, 1927. 



THE DOUBLE SLIT 



32S 



1 




ing the interference fringes, provided the width of the source is made 
correspondingly small. Just how small it must be is seen from the fact 
that the path difference from the extreme edges of the source to any given 
point on the screen such as P must be less than X/4. Now if we call s the 
width of the source, the discussion given in Sec. 15.11 shows that this 
path difference will be 2s sin 0/2. Hence, for a divergence of 60°, s can- 
not exceed one-quarter of a wavelength, or 1.3 X 10 -5 cm for green light. 
If the width is made greater than this the fringes disappear completely 
when the path difference is X, reappear, and then disappear again at 2X, 
etc., just as in the stellar interferometer. By using as a source an 

(6) 



Fluorescent 
layer 



Fig. 16/. Two methods of investigating wide-angle interference. 



extremely thin filament, Schrodinger could still detect some interference 
at an angular divergence <£ as large as 57°. 

An equivalent experiment which permitted using even larger angles of 
divergence (up to 180°) was performed by Selenyi in 1911. The essential 
part of his apparatus, shown in Fig. 167(6), was a film of a fluorescent 
liquid only ^X thick contained between a thin sheet of mica and a plane 
glass surface. When the film is strongly illuminated it becomes a second- 
ary source of light having a somewhat longer wavelength than the 
incident light (see Sec. 22.5, ahead). Interference may then be observed 
in a given direction between the light that comes directly from the film 
and that which is reflected from the outer surface of the mica. Inter- 
esting conclusions about the characteristics of the radiating atoms, in 
particular whether they radiate as dipoles, quadrupoles, etc., can be 
drawn from data on the variation of the visibility of the fringes with 
angle.* 




PROBLEMS 

1. Prove that Eq. 16c may be reduced, for the case where d = b, to the equation 
for the intensity distribution from a single slit of width 26. 

2. The widths of the individual slits of a double slit are each 0.17 mm, and the 
separation between their centers is 0.85 mm. Are there missing orders, and if so 

* O. Halpern and F. W. Doermann, Phys. Rev., 52, 937, 1937. 



326 PHYSICAL OPTICS 

which ones? What are the approximate relative intensities of the orders m = and 
m = 3? Ans. m = 5, 10, 15, . . . missing. 1:0.22. 

3. The double slit of Prob. 2 is illuminated by parallel light of wavelength 4358 A, 
and the pattern is focused on a screen by a lens of focal length 60 cm. Make a 
qualitative plot of the intensity distribution on the screen similar to Fig. 16D(c), 
but using as abscissas the distance in millimeters on the screen. Include the first 
14 orders on one side of the central maximum. 

4. Draw the appropriate vibration curve for the point where the phase difference 
5 = 2ir/3 in the case of a double slit where the opaque space between the slits is 
half the width of the slits themselves. What is the value of /3 for this point? Obtain 
graphically a value for the intensity at the point in question relative to that of the 
central maximum. Ans. /3 = 40°. / = 0.212 

6. Make a qualitative sketch of the intensity pattern produced by a double slit 
having d = 2.6b. Take the intensity of the central maximum as unity, and label 
the axis of abscissas with the values of m and p as in Fig. 16D(c). 

6. The Fraunhofer pattern from a double slit composed of slits each 0.5 mm wide 
and separated by d = 20 mm is observed in sodium light on a screen 50 cm behind 
the slits. Assuming that the eye can resolve fringes that subtend 1 minute of arc, 
what magnification would be required to see them in this case? How many fringes 
would occur under the central diffraction maximum? Under one of the side maxima? 

Ans. 4.9 X. 79. 39. 

7. Derive Eq. 16c by the use of the method of complex amplitudes described in 
Sec. 14.8. 

8. If d = 46 for a double slit, determine exactly how much the second-order maxi- 
mum is shifted from the position given by Eq. 16gr due to modulation by the diffraction 
envelope. The problem may best be solved by plotting the intensities in the neigh- 
borhood of the expected maximum. Express the result as a fraction of the separation 
of orders. Ans. 0.048 order toward center. 

9. On an optical bench are placed two double slits. The first has a slit spacing 
d = 0.2 mm and has a sodium arc placed immediately in front of it at one end of the 
bench. The eye, located close behind the second slit, for which d = 0.8 mm, sees 
clear double-slit fringes when observing at the far end of the bench. When the eye 
and the second double slit are moved together toward the source slits, the fringes 
completely disappear at a certain point. Find the largest distance between the 
double slits for this to occur. 

10. Calculate the value for the visibility (Sec. 13.12) of the double-slit fringes 
when the source consists of a double star separated by one-tenth the distance required 
for complete disappearance. This is the condition mentioned in Sec. 16.8 as being 
just perceptible to the eye. Ans. 98.8%. 

11. In making observations on the Fraunhofer pattern from a double slit with 
b = 0.12 mm, d = 0.78 mm, the latter is placed between two lenses as illustrated in 
Fig. 16G(a). The lenses have focal lengths of 85 cm. The source slit is illuminated 
by light of the green mercury fine. According to the usual criterion for clear fringes, 
how wide may the source slit be made to obtain the best intensity without appreciable 
sacrifice of clearness? 

12. Derive a formula giving the number of interference maxima occurring under 
the central diffraction maximum of the double-slit pattern, in terms of the slit separa- 
tion d and the slit width b. Ans. 2(d/b) — 1. 

13. The largest star measured by Michelson with his stellar interferometer was 
the red giant Betelgeuse. The effective wavelength of the light from this star may 
be taken as 5700 A. Complete disappearance of the fringes first occurred when tb<} 



THE DOUBLE SLIT 327 

mirrors were 10 ft 1 in apart. Compute the angular diameter in seconds of the star 
disk. 

14. Young first performed his famous experiment by observing interference in the 
light coming through two pinholes close together. Suppose these to be round and 
0.4 mm in diameter. For light of wavelength 5550 A, how close together would they 
have to be placed in order for the two Airy disks to overlap by one-half the radius of 
each when observed 1 m. behind the pinholes? Make a qualitative sketch of the 
pattern as it would appear to the eye. Ans. 2.54 mm. 3 interference fringes. 

15. With a single tungsten lamp filament as a source, and a collimating lens of 
3.5 cm focal length in front of a double slit, various separations of the double slit are 
tried, increasing them until the fringes no longer appear. If this occurs for d = 8 mm, 
estimate the diameter of the filament. Assume X = 6000 A. 

16. The interference fringes formed in Selenyi's experiment are evidently neither 
double-slit nor multiple-reflection fringes. To which of the various arrangements for 
producing interference described in Chaps. 13 and 14 is this one most closely related? 

Aiis. Michelson interferometer. 



CHAPTER 17 

THE DIFFRACTION GRATING 



Any arrangement which is equivalent in its action to a number of 
parallel equidistant slits of the same width is called a diffraction grating. 
Since the grating is a very powerful instrument for the study of spectra, 
we shall treat in considerable detail the intensity pattern which it pro- 
duces. We shall find that the pattern is quite complex in general but 
that it has a number of features in common with that of the double slit 
treated in the last chapter. In fact, the latter may be considered as an 
elementary grating of only two slits. It is, however, of no use as a 
spectroscope, since in a practical grating many thousands of very fine 
slits are usually required. The reason for this becomes apparent when 
we examine the difference between the pattern due to two slits and that 
due to many slits. 

17.1. Effect of Increasing the Number of Slits. When the intensity 
pattern due to one, two, three, and more slits of the same width is photo- 
graphed, a series of pictures like those shown in Fig. 17 A (a) to (/) is 
obtained. The arrangement of light source, slit, lenses, and recording 
plate used in taking these pictures was similar to that described in pre- 
vious chapters, and the light used was the blue line from a mercury arc. 
These patterns therefore are produced by Fraunhofer diffraction. In 
fact, it was because of Fraunhofer's original investigations of the diffrac- 
tion of parallel light by gratings in 1819 that his name became associated 
with this type of diffraction. Fraunhofer's first gratings were made by 
winding fine wires around two parallel screws. Those used in preparing 
Fig. 17 A were made by cutting narrow transparent lines in the gelatin 
emulsion on a photographic plate, as described in Sec. 13.2. 

The most striking modification in the pattern as the number of slits is 
increased consists of a narrowing of the interference maxima. For two 
slits these are diffuse, having an intensity which was shown in the last 
chapter to vary essentially as the square of the cosine. With more slits 
the sharpness of these principal maxima increases rapidly, and in pattern 
(/) of the figure, with 20 slits, they have become narrow lines. Another 
change, of less importance, which may be seen in patterns (c), (d), and (e) 
is the appearance of weak secondary maxima between the principal 

328 



THE DIFFRACTION GRATING 



329 






(c) rn 

Fi<;. 17/1. Fraunhofer diffraction patterns for gratings containing different numbers of 
slits. 

maxima, their number increasing with the number of slits. For three 
slits only one secondary maximum is present, its intensity being 11.1 per 
cent of the principal maximum. Figure 17 B shows an intensity curve 
for this case, plotted according to the theoretical equation 176 given 
in the next section. Here the individual slits were assumed very narrow. 




Sin — - 
Fig. 17 B. Principal and secondary maxima from a grating of three slits. 

Actually the intensities of all maxima are governed by the pattern of a 
single slit of width equal to that of any one of the slits used. The width 
of the intensity envelopes would be identical in the various patterns of 
Fig. 17A if the slits had been of the same width in all cases. In fact 
there were slight differences in the slit widths used for some of the 
patterns. 



330 PHYSICAL OPTICS 

17.2. Intensity Distribution from an Ideal Grating. The procedure 
used in Sees. 15.2 and 16.2 for the single and double slits could be used 
here, performing the integration over the clear aperture of the slits, but 
it becomes somewhat cumbersome. Instead let us apply the more power- 
ful method of adding the complex amplitudes (Sec. 14.8). The situation 
is simpler than in the case of multiple reflections, because for the grating 
the amplitudes contributed by the individual slits are all of equal magni- 
tude. We designate this magnitude by a, and the number of slits by N. 
The phase will change by equal amounts 5 from one slit to the next; so 
the resultant complex amplitude is the sum of the series 

Ae ie = a(l -f e a + e i2S + e m + • • • + e i( - v ~ 1)6 ) 

I _ e iNS 

To find the intensity, this expression must be multiplied by its complex 
conjugate, as in Eq. 14*, giving 

(1 - e«)(l - e-«) 

„ 1 — cos N8 

= a ~i T" 

1 — cos 5 

Using the trigonometric relation 1 — cos a = 2 sin 2 (a/2), we may then 
write 

= fl2 si" 2 (M/2) = sin*JV 7 

° sin 2 (5/2) ° sin 2 7 ( } 

where, as in the double slit, 7 = 5/2 = (vd sin 0)/X. Now the factor a 2 
represents the intensity diffracted by a single slit, and after inserting its 
value from Eq. I5d we finally obtain for the intensity in the Fraunhofer 
pattern of an ideal grating 

T ., . , sin 2 /3 sin 2 JV7 

I ~ A 2 = io 8 -^ . . (17c) 

/3 2 sin 2 7 v ' 

Upon substitution of N = 2 in this formula, it readily reduces to Eq. 16c 
for the double slit. 

17.3. Principal Maxima. The new factor (sin 2 #7)/ (sin 2 7) may be 
said to represent the interference term for AT" slits. It possesses maximum 
values equal to N 2 for 7 = 0, r, 2w, . . . . Although the quotient 
becomes indeterminate at these values, this result may be obtained by 
noting that 



lim (sJIlJfy) = Um (N_S^Ny\ = 
->-.*.* \ sin 7 / y -, mx \ cos 7 / 



= ±tf 



THE DIFFRACTION GRATING 331 

These maxima correspond in position to those of the double slit, since 
for the above values of 7 

J • n r» -» fw ox x PRINCIPAL .,_ ,. 

d sin 6 = 0, X, 2X, 3X, . . . = raX (17d) 

MAXIMA 

They are more intense, however, in the ratio of the square of the number 
of slits. The relative intensities of the different orders m are in all cases 
governed by the single-slit diffraction envelope (sin 2 /3)//3 2 . Hence the 
relation between /3 and 7 in terms of slit width and slit separation (Eq. 
16d) remains unchanged, as does the condition for missing orders (Eq. 
16/i). 

17.4. Minima and Secondary Maxima. To find the minima of the 
function (sin 2 A^7)/(sin 2 7), we note that the numerator becomes zero 
more often than the denominator, and this occurs at the values Ny = 0, 
7r, 2ir, . . . or, in general, pir. In the special cases when p = 0, N, 
2N, . . . , 7 will be 0, x, 2ar, . . . ; so for these values the denominatoi \ 
will also vanish, and we have the principal maxima described above. 
The other values of p give zero intensity, since for these the denominator 
does not vanish at the same time. Hence the condition for a minimum 
is 7 = pw/N, excluding those values of p for which p = mN, m being 
the order. These values of 7 correspond to path differences 

, . . X 2X 3X (N - 1)X (A r + 1)X 
dsmd = N'N'N' ' • • > If ' N ' • • ■ mNIMA (17e) 

omitting the values 0, N\/N, 2NX/N, . . . , for which d sin 6 = mX 
and which according to Eq. 17d represent principal maxima. Between 
two adjacent principal maxima there will hence be N — I points of zero 
intensity. The two minima on either side of a principal maximum are 
separated by twice the distance of the others. 

Between the other minima the intensity rises again, but the secondary 
maxima thus produced are of much smaller intensity than the principal 
maxima. Figure 17C shows a plot for six slits of the quantities sin 2 JV7 
and sin 2 7, and also of their quotient, which gives the intensity distri- 
bution in the interference pattern. The intensity of the principal max- 
ima is N 2 or 36, so that the lower figure is drawn to a smaller scale. The 
intensities of the secondary maxima are also shown. These secondary 
maxima are not of equal intensity but fall off as we go out on either side 
of each principal maximum. Nor are they in general equally spaced, 
the lack of equality being due to the fact that the maxima are not quite 
symmetrical. This lack of symmetry is greatest for the secondary max- 
ima immediately adjacent to the principal maxima, and is such that the 
secondary maxima are slightly shifted toward the adjacent principal 
maximum. 



332 



PHYSICAL OPTICS 



These features of the secondary maxima show a strong resemblance 
to those of the secondary maxima in the single-slit pattern. Comparison 
of the central part of the intensity pattern in Fig. 17C(d) with Fig. 15Z> 
for the single slit will emphasize this resemblance. As the number of 




1.0 

SinW-r 

CL 

7=0 



(b) 




7T 2n 3jt 47t 5jt 6n 
%2%3%4%5% ft 






Sin0=O 

Fig. 17C. Fraunhofer diffraction by a grating of six very narrow slits, and 
the intensity pattern. 



|ff=20l 



"tfc ftfs T y)k\ n 



y^hr 



N7 = n2irZn 18»rl9jr 21n 22n 38n 39n 41* 42n 58tt 59rr 6\n 62tt 
Fig. 17i>. Intensity pattern for 20 narrow slits. 

slits is increased, the number of secondary maxima is also increased, 
since it is equal to N — 2. At the same time the resemblance of any 
principal maximum and its adjacent secondary maxima to the single-slit 
pattern increases. In Fig. 17 D is shown the interference curve for 
N = 20, corresponding to the last photograph shown in Fig. 17 A. In 
this case there are 18 secondary maxima between each pair of principal 
maxima, but only those fairly close to the principal maxima appear with 
appreciable intensity, and even these are not sufficiently strong to show 
in the photograph. The agreement with the single-slit pattern is here 



THE DIFFRACTION GRATING 



333 



practically complete. The physical reason for this agreement will be 
discussed in Sec. 17.10, where it will be shown that the dimensions of the 
pattern correspond to those from a single "slit" of width equal to that 
of the entire grating. Even when the number of slits is small it may be 
shown that the intensities of the secondary maxima can be computed by 
summing a number of such single-slit patterns, one for each order (Prob. 
6). 




Fig. 172?. Positions and intensities of the principal maxima from a grating, where light 
containing two wavelengths is incident at an angle i and diffracted at various angles 0. 

17.6. Formation of Spectra by a Grating. The secondary maxima 
discussed above are of little importance in the production of spectra by 
a many-lined grating. The principal maxima treated in Sec. 17.3 are 
called spectrum lines because when the primary source of light is a narrow 
slit they become sharp, bright lines on the screen. These lines will be 
parallel to the rulings of the grating if the slit also has this direction. 
For monochromatic light of wavelength X, the angles 6 at which these 
lines are formed are given by Eq. 17d, which is the ordinary grating 
equation d sin 6 = mX commonly given in elementary textbooks. A more 
general equation includes the possibility of light incident on the grating 
at any angle i. The equation then becomes 



d(sin i + sin 6) = raX grating equation 



(17/) 



since, as will be seen from Fig. 172?, this is the path difference for light 



334 



PHYSICAL OPTICS 



passing through adjacent slits. The figure shows the path of thr? light 
forming the maxima of order m = (called the central image), and also 
m = 4 in light of a particular wavelength Xi. For the central image, 
Eq. 17/ shows that sin = — sin i, or 6 = —i. The negative sign 
comes from the fact that we have chosen to call i and positive when 
measured on the same side of the normal; i.e., our convention of signs 
is such that whenever the rays used cross over the line normal to the 
grating, is taken as negative. Those intensity maxima which are 





-4 -3 -2 -l CI. 
Fig. 17F. Grating spectra of two wavelengths, 
(c) Xi and Xj together. 



(a) Xi 



a 4 

4000 L (b) X 2 



5000 A. 



shaded show the various orders of the wavelength Xi. In the case of 
the fourth order, for example, the path differences indicated are such that 
d(sin i -f- sin 0) = 4X X . The intensities of the principal maxima are lim- 
ited by the diffraction pattern corresponding to a single slit (broken line) 
and drop to zero at the first minimum of that pattern, which here coin- 
cides with the fifth order. The missing orders in this illustration are 
therefore m = 5, 10, . . . , as would be produced by having d = ob. 

Now if the source gives light of another wavelength X2 somewhat 
greater than Xi, the maxima of the corresponding order m for this wave- 
length will, according to Eq. 17/, occur at larger angles 6. Since the 
spectrum lines are narrow, these maxima will in general be entirely 
separate in each order from those of Xi, and we have two lines forming a 
line spectrum in each order. These spectra are indicated by brackets 
in the figure. Both the wavelengths will coincide, however, for the 
central image, because for this the path difference is zero for any wave- 
length. A similar set of spectra occurs on the other side of the central 



THE DIFFRACTION GRATING 335 

image, the shorter wavelength line in each order lying on the side toward 
the central image. Figure 17F shows actual photographs of grating 
spectra corresponding to the diagram of Fig. 17 E. If the source gives 
white light, the central image will be white, but for the orders each 
will be spread out into continuous spectra composed of an infinite number 
of adjacent images of the slit in light of the different wavelengths present. 
At any given point in such a continuous spectrum, the light will be very 
nearly monochromatic because of the narrowness of the slit images 
formed by the grating and lens. The result is in this respect funda- 
mentally different from that with the double slit, where the images were 
broad and the spectral colors were not separated. 

17.6. Dispersion. The separation of any two colors, such as Xi and X 2 
in Figs. 17E and 17^, increases with the order number. To express this 
separation the quantity frequently used is called the angular dispersion, 
which is denned as the rate of change of angle with change of wavelength. 
An expression for this quantity is obtained by differentiating Eq. 17/ 
with respect to X, remembering that i is a constant independent of wave- 
length. Substituting the ratio of finite increments for the derivative, 
one has 

— - = , m n ANGULAR DISPERSION (170) 

AX a cos 6 

The equation shows in the first place that for a given small wavelength 
difference AX, the angular separation Ad is directly proportional to the 
order m. Hence the second-order spectrum is twice as wide as the first 
order, the third three times as wide as the first, etc. In the second place, 
A0 is inversely proportional to the slit separation d, which is usually 
referred to as the grating space. The smaller the grating space, the more 
widely spread will be the spectra. In the third place, the occurrence of 
cos in the denominator means that in a given order m the dispersion 
will be smallest on the normal, where 0=0, and will increase slowly as 
we go out on either side of this. If does not become large, cos will 
not differ much from unity, and this factor will be of little importance. 
If we neglect its influence, the different spectral lines in one order will 
differ in angle by amounts which are directly proportional to their differ- 
ence in wavelength. Such a spectrum is called a normal spectrum, and 
one of the chief advantages of gratings over prism instruments is this 
simple linear scale for wavelengths in their spectra. 

The linear dispersion in the focal plane of the telescope or camera 
lens is AJ/AX, where I is the distance along this plane. Its value is usually 
obtainable by multiplying Eq. 17# by the focal length of the lens. In 
some arrangements, however, the photographic plate is turned so the 
light does not strike it normally, and there is a corresponding increase in 



336 PHYSICAL OPTICS 

linear dispersion. In specifying the dispersion of a spectrograph, it has 
become customary to quote the so-called plate facto?', which is the recip- 
rocal of the above quantity and is expressed in angstroms per millimeter. 

17.7. Overlapping of Orders. If the range of wavelengths is large, 
for instance, if we observe the whole visible spectrum between 4000 and 
7200 A, considerable overlapping occurs in the higher orders. Suppose, 
for example, that one observed in the third order a certain red line of 
wavelength 7000 A. The angle of diffraction for this line is given by 
solving for 6 the expression 

d(sm i + sin 6) = 3 X 7000 

where d is in angstrom units. But at the same angle there may occur 
a green line in the fourth order, of wavelength 5250 A, since 

4 X 5250 = 3 X 7000 

Similarly the violet of wavelength 4200 A will occur in the fifth order 
at this same place. The general condition for the various wavelengths 
that can occur at a given angle is then 

d(sin i + sin 8) - Xi = 2X 2 = 3A 3 = • • • (I7h) 

where Xi, X2, etc., are the wavelengths in the first, second, etc., orders. 
For visible light there is no overlapping of the first and second orders, 
since with Xi = 7200 A and X 2 = 4000 A the red end of the first order 
falls just short of the violet end of the second. When photographic 
observations are made, however, these orders may extend down to 2000 A 
in the ultraviolet, and the first two orders do overlap. This difficulty 
may usually be eliminated by the use of suitable color filters to absorb 
from the incident light those wavelengths which would overlap the region 
under study. As an example, a piece of red glass transmitting only 
wavelengths longer than 6000 A could be used in the above case to avoid 
the interfering shorter wavelengths of higher order that might disturb 
observation of X7000 and lines in that vicinity. 

17.8. Width of the Principal Maxima. It was shown at the beginning 
of Sec. 17.4 that the first minima on either side of any principal maximum 
occur where Ny = mNir ± it, or where 7 = mir ± (ir/N) . When 7 = tut, 
we have the principal maxima, owing to the fact that the phase difference 
8 or 27, in the light from corresponding points of adjacent slits, is given by 
2irm, or a whole number of complete vibrations. However, if we change 
the angle enough to cause a change of 2ir/N in the phase difference, 
reinforcement no longer occurs, but the light from the various slits now 
interferes to produce zero intensity. A phase difference of 2ir/N between 
the maximum and the first minimum means a path difference of A/AT. 



THE DIFFRACTION GRATING 



337 



To see why this path difference causes zero intensity, consider Fig. 
\7G(a), in which the rays leaving the grating at the angle 6 form a 
principal maximum of order m. For these, the path difference of the 
rays from two adjacent slits is raX, so that all the waves arrive in phase. 
The path difference of the extreme rays is then Nm\, since N is always 
a very large number in any practical case.* Now let us change the 
angle of diffraction by a small amount Ad, such that the extreme path 
difference increases by one wavelength and becomes Nm\ + X (rays 



Nd< 




Fig. 17G. Angular separation of two spectrum lines which are just resolved by a 
diffraction grating. 

shown by broken lines). This should correspond to the condition for 
zero intensity, because as is required the path difference for two adjacent 
slits has been increased by X/2V. It will be seen that the ray from the 
top of the grating is now of opposite phase from that at the center, and 
the effects of these two will cancel. Similarly, the ray from the next 
slit below the center will annul that from the next slit below the top, etc. 
The cancellation if continued will yield zero intensity from the whole 
grating, in entire analogy to the similar process considered in Sec. 15.3 
for the single-slit pattern. 

Thus the first zero occurs at the small angle A0 on each side of any 
principal maximum. From the figure, it is seen that 



A6 = * = x 

B Nd cos 



ANGULAR HALF WIDTH OF PRINCIPAL 
MAXIMUM 



(170 



* With a small number of slits, it is necessary to use the true value (N — l)mX, 
and the subsequent argument must be slightly modified, but yields the same result 
(Eq. 17*). 



338 



PHYSICAL OPTICS 



It is instructive to note that this is just 1/Nth of the separation of oxtjatenl 
orders, since the latter is represented by the same expression with the 
path difference NX instead of X in the numerator. 

17.9. Resolving Power. When N is many thousands, as in any useful 
diffraction grating, the maxima are extremely narrow. The chromatic 
resolving power X/AX is correspondingly high. To evaluate it, we note 
first that since the intensity contour is essentially the diffraction pattern 
of a rectangular aperture, the Rayleigh criterion (Sec. 15.6) may be 
applied. The images formed in two wavelengths that are barely resolved 
must be separated by the angle A0 of Eq. 17*. Consequently the light 
of wavelength X + AX must form its principal maximum of order m at 
the same angle as that for the first minimum of wavelength X in that 
order [Fig. 17Cr(6)]. Hence we may equate the extreme path differences 
in the two cases, and obtain 

mNX + X = mN(\ + AX) 

from which it immediately follows that 

s - mN ow 

That the resolving power is proportional to the order m is to be under- 
stood from the fact that the width of a principal maximum, by Eq. 17i, 
depends on the width B of the emergent beam and does not change much 
with order, whereas the separation of two maxima of different wave- 
lengths increases with the dispersion, which, by Eq. 17g, increases nearly 
in proportion to the order. Just as for the prism (Sec. 15.7), we have 
that 

Chromatic resolving power = angular dispersion 

X width of emergent beam 
since in the present case 

-^ = ^XB = T^-a X Nd cos - mN 
AX AX d cos 

In a given order the resolving power, by Eq. 17 j, is proportional to the 
total number of slits N, but is independent of their spacing d. However, 
at given angles of incidence and diffraction it is independent of N also, 
as can be seen by substituting in Eq. 17,;' the value of m from Eq. 17/. 

X _ d(sin i + sin 0) „ _ TT(sin i -f- sin 0) . _ . 

AX = X " ~ ~ X ( Uk) 

Here W = Nd is the total width of the grating. At a given i and 0, 
the resolving power is therefore independent of the number of lines ruled 



THE DIFFRACTION" GRATIXG 



339 



in the distance W. A grating with fewer lines gives a higher order at 
these given angles, however, with consequent overlapping, and would 
require some auxiliary dispersion to separate these orders, as does the 
Fabry-Perot interferometer. The method has nevertheless been recently 
applied with success in the echelle grating to be described later. Theo- 
retically the maximum resolving power obtainable with any grating 




d=Zb 



$fc*itm*^rmr*jfflhi± 




izx 



*rm, dv* «^~s / ;:; 




(a) 



(6) 






a b c d e f 

^3 2^ 7T 4% 5^/3 2* 

% 



At 



^2_ _A 3 A 4 

* * > > > 



3* 



4tt 
2% 



A 6 



A=A, 



*-%\ 



= */- 



V 



H^i 



A, 



./ 



A=0 
/ A t A 6 \ 



y= 



» \ A 4 / 



U) 






A, 



(«) 



\ A 3 A 4 / 



A=0 



\/ 



(/) 



o 



A=0 



^^/ 12 



5^ 



-27T, 



Fig. 177/. Illustrating how the intensity curve for a grating of several slits is obtained 
by the graphical addition of amplitudes. 

occurs when i = 6 = 90°, and according to Eq. 17/c it equals 2W/X, or 
the number of wavelengths is twice the width of the grating. In prac- 
tice such grazing angles are not usable, however, because of the negligible 
amount of light. One can only hope to reach about two-thirds of the 
ideal maximum. 

17.10. Vibration Curve. Let us now apply the method of compound- 
ing the amplitudes vectorially which was used in Sec. 16.6 for two slits 
and in Sec. 15.4 for one slit. The vibration curve for the contributions 
from the various infinitesimal elements of a single slit again forms an 
arc of a circle, but there are now several of these arcs in the curve, 
corresponding to the several slits of the grating. In Fig. 17 H the 



340 



PHYSICAL OPTICS 



diagrams corresponding to the various points (a) to (/) of the intensity 
plot for six slits are shown. For the central maximum the light from 
all slits, and from all parts of each slit, is in phase, giving a resultant 
amplitude A which is N times as great as that from one slit, as shown in 
(a) of the figure Halfway to the first minimum the condition is as 
shown in (6). For this point y = tt/12, so that the phase difference 
from corresponding points in adjacent slits 5 equals 7r/6 (cf. Fig. 17C). 

This is also the angle between 
successive vectors in the series of 
six resultants A\ to A 6 which are 
the chords of six small equal arcs. 
Just as for the double slit, the final 
resultant A is obtained by com- 
pounding these vectorially, and 
the intensity is measured by A 2 . 
With increasing angle the individ- 
ual resultants become slightly 
smaller in magnitude as in- 
creases, because it is the arc, not the chord, which is constant in length. 
Their difference is here small, even for point (/). 

The derivation of the general intensity function for the grating, Eq. 
176, can be very simply done by a geometrical method. In Fig. 17/ 
the six amplitude vectors of Fig. 17 H are shown with a phase difference 
somewhat less than in part (6) of the figure. All these have the same 
magnitude, given by 




A 3 A 4 

Fig. 17/. Geometrical derivation of the 
intensity function for a grating. 



A n = S ^A 



(171) 



since this represents the chord of an arc of length A subtending the 
angle 2/3 (see Fig. 15F). Each vector is inclined to the next by the 
angle 8 = 2t, and thus the six form part of a regular polygon. In the 
figure broken lines are drawn from the ends of each vector to the center 
of this polygon. These lines also make the constant angle 27 with each 
other. Therefore the total angle subtended at the center is 

4> = Nb = N X 27 

We wish the relation between the resultant amplitude A and the indi- 
vidual ones A n , which are given by Eq. 17/. By dividing the triangle 
OBC into two halves with a line from perpendicular to A, it is seen that 

<i> 
A — 2r sin =r 



where r represents OB or OC. Similarly, from the triangle OBD as 



THE DIFFRACTION GRATING 341 

split by a line perpendicular to Ai, we obtain 

A n = A\ — 2r sin 7 
Dividing this equation into the previous one, we find 



. 2r sin s . ,, 

A _2 _ sin N y 

An 2r sin 7 sin 7 

When we then substitute the value of A „ from Eq. 17/, there results, 
for the amplitude, 

sin sin Ny 



A = A, 



sin 7 



The square of this, which gives the intensity, is seen to be identical with 
Eq. 17c. 

The vibration curve as applied to different numbers of slits helps to 
understand many features of the intensity patterns. For instance, there 
is the important question of the narrowness of the principal maxima. 
The adjacent minimum on one side is reached when the vectors first 
form a closed polygon, as is (c) of Fig. 17//. It is evident that this will 
occur for smaller values of 8 the larger the number of slits, and this means 
that the maxima will become sharper. Also one can see at once from 
the diagram that for this minimum 5 = 2-r/N, or 7 = ir/N, the condition 
stated at the beginning of Sec. 17.8. Furthermore, as the number of 
slits becomes large, the polygon of vectors will rapidly approach the 
arc of a circle, and the analogy with the pattern due to a single aperture 
of width equal to that of the grating is thereby seen to be justified. 
Comparison of Fig. 17 H with Fig. 15F for the single slit will show that 
for large N the diagrams for the grating will become identical with those 
for one slit if we replace N8/2 or Ny by 0. Since Ny is half the phase 
difference from extreme slits of the grating and half the phase difference 
between extreme points in an open aperture, we see the physical reason 
for the correspondence mentioned in Sec. 17.4. 

Finally we note that if the diagrams in Fig. 17// are carried further, 
the first-order principal maximum occurs when the arc representing each 
interval d forms one complete circle. The chords under these conditions 
are all parallel and in the same direction as in (a), but smaller in magni- 
tude. The second principal maximum occurs when each arc forms two 
turns of a circle when the resultant chords again line up. These maxima 
have no analogue in the pattern for a single slit. 

17.11. Production of Ruled Gratings. Up to this point we have con- 
sidered the characteristics of an idealized grating consisting of identical 
and equally spaced slits separated by opaque strips. Actual gratings 



342 PHYSICAL OPTICS 

used in the study of spectra are made by ruling fine grooves with a dia- 
mond point either on a plane glass surface to produce a transmission grat- 
ing or more often on a polished metal mirror to produce a reflection grating. 
The transmission grating gives something like our idealized picture, 
since the grooves scatter the light and are effectively opaque, while the 
undisturbed parts of the surface transmit regularly and act like slits. 
The same is true of the reflection grating, except that here the unruled 
portions reflect regularly, and the grating equation 17/ holds equally 
well for this case with the same convention of signs for i and 0. 




(a) (6) 

Fig. 17/. Microphotographs of the rulings on reflection gratings, (a) Light ruling, 
(ft) Heavy ruling. {After H. D. Babcock.) 

Figure 17 J shows microphotographs of the ruled surfaces of two differ- 
ent reflection gratings. The grating shown in (a) was ruled lightly, and 
the grooves are too shallow to obtain maximum brightness. That shown 
in (6) was a high-quality grating having 15,000 lines per inch. One 
or two vertical cross-rulings have been made to show more clearly the 
contour of the ruled surface. 

Until recently, most gratings were ruled on speculum metal, a very 
hard alloy of copper and tin. Modern practice, however, is to rule on 
an evaporated layer of the softer metal aluminum. Not only does this 
give greater reflection in the ultraviolet, but it causes less wear on the 
diamond ruling point. The chief requirement for a good grating is that 
the lines shall be as nearly equally spaced as possible over the whole 
ruled surface, which in different gratings varies from 1 to 10 in. in width. 
This is a difficult requirement to fulfill, and there are very few places 
in the world where ruling machines of precision adequate for the produc- 
tion of fine gratings have been constructed. After each groove has been 
ruled, the machine lifts the diamond point and moves the grating forward 



THE DIFFRACTION GRATING 343 

by a small rotation of the screw which drives the carriage carrying it. 
To have the spacing of rulings constant, the screw must be of very 
constant pitch, and it was not until the manufacture of a nearly perfect 
screw had been achieved by Rowland,* in 1882, that the problem of 
successfully ruling large gratings was accomplished. 

If ruled gratings are used without any auxiliary apparatus to separate 
the different orders, the overlapping of these makes it impractical to use 
values of m above 4 or 5. Hence, to obtain adequate dispersion and 
resolving power, the grating space must under these circumstances be 
made very small, and a large number of lines must be ruled. Rowland's 
engine gave 14,438 per inch, corresponding to d = 1.693 X 10~ 4 cm, and 
could produce gratings nearly 6 in. wide. This grating space is about 
three wavelengths of yellow light, and thus the third order is the highest 
that can be observed in this color with normal incidence. Correspond- 
ingly higher orders can be observed for shorter wavelengths. Even in 
the first order, however, the dispersion given by such a grating far exceeds 
that of a prism. From the grating equation one finds that the visible 
spectrum is spread over an angle of 12°. If it were projected by a lens 
of 3 m focal length, the spectrum would cover a length of about 60 cm 
on the photographic plate. In the second order it would be more than a 
meter long. 

The real advantage of the grating over the prism lies not in its large 
dispersion, however, but in the high resolving power it affords. One 
can always increase the linear dispersion by using a camera lens of longer 
focal length, but beyond a certain minimum set by the graininess of the 
photographic plate no more detail is revealed thereby. With sufficient 
dispersion, the final limitation is the chromatic resolving power. A 6-in. 
Rowland grating in the first order gives X/AX = 6 X 14,438 c~ 76,600. 
In the orange region two lines only 0.08 A apart would be resolved, and 
with the above-mentioned dispersion each line would be only 0.015 mm 
wide. This separation is only one-eightieth of that of the orange sodium 
doublet. A glass prism, even though it had the rather large dn/dk 
of — 1200 cm -1 , would by Eq. \bh need to have a base 64 cm long to yield 
the same resolution. 

It was first shown by Thorp that fairly good transmission gratings 
could be made by taking a cast of the ruled surface with some transpar- 
ent material. Such casts are called replica gratings, and may give satis- 
factory performance where the highest resolving power is not needed. 
Collodion or cellulose acetate, properly diluted, is poured on the grating 

* II. A. Rowland (1848-1901). Professor of physics at the Johns Hopkins Uni- 
versity, Baltimore. He is famous for his demonstration of the magnetic effect of a 
charge in motion, for his measurements of the mechanical equivalent of heat, and for 
his invention of the concave grating (Sec. 17.15). 



344 PHYSICAL OPTICS 

surface and dried to a thin, tough film which can easily be detached from 
the master grating under water. It can then be mounted on a plane 
glass plate or concave mirror. Some distortion and shrinkage is involved 
in this process, so that the replica seldom functions as well as the master. 
With modern improvements in the techniques of plastics, however, 
replicas of high quality are now being made. 

17.12. Ghosts. In an actual grating the ruled linos will always 
deviate to some extent from the ideal of equal spacing. This gives rise 
to various effects, according to the nature of the ruling error. Three 
types may be distinguished. (I) The error is perfectly random in mag- 
nitude and direction. In this case the grating will give a continuous 
spread of light underlying the principal maxima, even when monochro- 
matic light is used. (2) The error continuously increases in one direction. 
This can be shown to give the grating "focal properties." Parallel light 
after diffraction is no longer parallel, but slightly divergent or convergent. 
(3) The error is periodic across the surface of the grating. This is the 
most common type, since it frequently arises from defects in the driving 
mechanism of the ruling machine. It gives rise to "ghosts," or false 
lines, accompanying every principal maximum of the ideal grating. When 
there is only one period involved in the error, these lines are symmetrical 
in spacing and intensity about the principal maxima. Such ghosts are 
called Rowland ghosts, and may easily be seen in Fig. 21 H(g). More 
troublesome, though of less frequent occurrence, are the Lyman* ghosts. 
These appear when the error involves two periods that are incommen- 
surate with each other, or else when there is a single error of very short 
period. Lyman ghosts may occur very far from the principal maximum 
of the same wavelength. 

17.13. Control of the Intensity Distribution among Orders. The rela- 
tive intensities of the different orders for a ruled grating do not conform 
to the term (sin 2 /3)//3 2 derived for the ideal case (Eq. 17c). Obviously 
the light reflected from (or refracted by) the sides of the grooves will 
produce important modifications. In general there will be no missing 
orders. The positions of the spectral lines are uninfluenced, however, 
and remain unchanged for any grating of the same grating space d. In 
fact, the only essential requirement for a grating is that it impress on 
the diffracted wave some periodic variation of either amplitude or phase. 
The relative intensity of different orders is then determined by the 
angular distribution of the fight diffracted by a single element, of width 
d, on the grating surface. In the ideal grating this corresponds to the 
diffraction from a single slit. In ruled gratings it will usually be a 

* Theodore Lyman (1874-1954). For many years director of the Physical Labora- 
tories at Harvard University. Pioneer in the investigation of the far ultraviolet 
spectrum. 



THE DIFFRACTION GRATING 



345 



complex factor, which in the early days of grating manufacture was con- 
sidered to bo largely uncontrollable. More recently, R. W. Wood has 
been able to produce gratings which concentrate as much as 90 per cent 
of the light of a particular wavelength in a single order on one side. Thus 
one of the chief disadvantages of gratings as compared to prisms — the 
presence of multiple spectra, none of which is very intense — is overcome. 
Wood's first experiments were done with gratings for the infrared, 
which have a large grating space so that the form of the grooves could 
be easily governed. These so-called echelette gratings had grooves 
with one optically flat side inclined at such an angle <f> as to reflect the 




e 


--o° 
















**-* 













ia) 



ib) 



Fi<;. 17A\ Concentration of light in a particular direction by (a) an echelette or 
echelle grating and (6) a reflection echelon. 

major portion of the infrared radiation toward the order that was to be 
bright [Fig. 17K(a)]. Of course the light from any one such face is 
diffracted through an appreciable angle, measured by the ratio of the 
wavelength to the width b of the face. When the ruling of gratings oh 
aluminum was started, it was found possible to control the shape of the 
finer grooves required for visible and ultraviolet light. By proper shap- 
ing and orientation of the diamond ruling point, gratings are now produced 
which show a blaze of light at any desired angle. 

Historically, the first application of the principle of concentrating the 
light in particular orders was made by Michelson in his echelon grating 
[Fig. 17 K(b)]. This instrument consists of 20 to 30 plane-parallel plates 
stacked together with a constant offset b of about 1 mm. The thickness 
I was usually 1 cm so that the grating space is very large and concentration 
occurs in an extremely high order. As used by Michelson, echelons were 
transmission instruments, but larger path differences and higher orders 
are afforded by the reflection type first made by Williams.* In either 
case, the light is concentrated in a direction perpendicular to the fronts 
of the steps. At most two orders of a given wavelength appear under the 
diffraction maximum. These have such large values of m [about 2t/\ 
for the reflection type and (n — l)t/\ for the transmission type] that 

* W. E. Williams, Proc. Phys. Soc. (London), 45, 099, 1933- 



346 



PHYSICAL OPTICS 



the resolving power mN is very high, even with a relatively small number 
N of plates. In this respect the instrument is like an interferometer, and 
in the same way requires auxiliary dispersion to separate the lines that 
are being studied. Since it has the same defect of lack of flexibility as 
does the Lummer-Gehrcke plate, the echelon is little used nowadays. 

A more important type of grating called the echelle, which is inter- 
mediate between the echelette and the echelon, has recently been devel- 



X5461 



1 




Fig. 17L. Echellegram of the thorium spectrum (After Sumner P. Davis). 

oped.* It has a relatively coarse spacing of the grooves, some 200 to the 
inch. These are shaped as in Fig. \lK(a), but with a rather steeper slope. 
The order numbers for which concentration occurs are in the hundreds, 
whereas for an echelon they are in the tens of thousands. An echelle 
must be used in conjunction with another dispersing instrument, usually 
a prism spectrograph, to separate the various orders. If the dispersion 
of the echelle is in a direction perpendicular to that of the prism, an 
extended spectrum is displayed as a series of short strips representing 
adjacent orders, as shown in Fig. 17L.f This is part of a more extensive 
spectrogram, which covers a large wavelength range with a plate factor 

* G. R. Harrison, J. Opt. Soc. Am., 39, 522, 1949; 43, 853, 1953. 

t The separation of orders, in taking the echellegram of Fig. 17L, was accomplished 
not by a prism but by an ordinary grating. This accounts for the weaker spectra 
between the orders marked, which occur in its second order and have echelle orders 
twice as great. 



THE DIFFRACTION GRATING 347 

of only 0.5 A/mm. Each order contains about 14 A of the spectrum, the 
range that is covered by the diffraction envelope of a single groove. Phis 
range is sufficient to produce a certain amount of repetition from one 
order to the next. Thus in Fig. 17L the green mercury line, which has 
been superimposed as a reference wavelength, appears in the 405th order, 
and again at the extreme left in the 404th order. The resolving powei 
afforded by the echelle depends only on its total width (Eq. 17k) and 
can be some fifty times higher than that of the auxiliary spectrograph. 
Here it is sufficient to resolve the hyperfine structure of the green line. 
Besides its high resolution and dispersion, the echelle has the advantages 
of yielding bright spectra, and of registering the spectra in very compact 
form. 

17.14. Measurement of Wavelength with the Grating. Small gratings 
1 or 2 in. wide are usually mounted on the prism table of a small spectrom- 
eter with collimator and telescope. By measuring the angles of incidence 
and diffraction for a given spectrum line its wavelength may be calculated 
from the grating formula (Eq. 17/). For this the grating space d needs 
to be known, and this is usually furnished with the grating. The first 
accurate wavelengths were determined by this method, the grating space 
being found by counting the lines in a given distance with a traveling 
microscope. Once the absolute wavelength of a single line is known, 
others may be measured relative to it by using the overlapping of orders. 
For instance, according to Eq. Ylh a sodium line of wavelength 5890 A 
in the third order will coincide with another line of A = f X 5890 = 4417 A 
in the fourth order. Of course no two lines will exactly coincide in this way, 
but they may fall close enough together so that the small difference can be 
accurately corrected for. This method of comparing wavelengths is not 
accurate with the arrangement described above, because the telescope 
lens is never perfectly achromatic and the two lines will not be focused 
in exactly the same plane. To avoid this difficulty Rowland invented 
the concave grating, in which the focusing is done by a concave mirror, 
upon which the grating itself is ruled. 

17.15. Concave Grating. If the grating, instead of being ruled on a 
plane surface, is ruled on a concave spherical mirror of metal, it will 
diffract and focus the light at the same time, thus doing away with the 
necessity of using lenses. Beside the fact that this eliminates the chro- 
matic aberration mentioned above, it has the great advantage that the 
grating may be used for regions of the spectrum which are not trans- 
mitted by glass lenses, such as the ultraviolet. A mathematical treat- 
ment of the action of the concave grating would be out of place here, 
but we may mention one of the more important results. It is found 
that if R is the radius of curvature of the spherical surface of the grating, 
a circle of diameter R (i.e., radius t = R/2) may be drawn tangent to 



348 



PHYSICAL OPTICS 



the grating at its mid-point which defines the locus of points where the 
spectrum is in focus, provided the source slit also lies on this circle. 
This circle is called the Rowland circle, and in practically all mountings 
for concave gratings use is made of this condition for focus. 

17.16. Grating Spectrographs. Figure 17 M shows a diagram of a 
common form of mounting used for large concave gratings, called the 
Paschen mounting. The slit is set up on the Rowland circle, and the 
light from this strikes the grating, which diffracts it into the spectra of 



Third order 
spectrum 



Grating 




Second order 
spectrum 



First order 
spectrum 



Central image 
Fig. 17M . Paschen mounting for a concave grating. 



various orders. These spectra will be in focus on the circle, and the 
photographic plates are mounted in a plate holder which bends them to 
coincide with this curve. Several orders of a spectrum can be photo- 
graphed at the same time in this mounting. The ranges covered by the 
visible spectrum in the first three orders are indicated in Fig. VIM for 
the value of the grating space mentioned above. In a given order, 
Eq. 17<7 shows that the dispersion is a minimum on the normal to the 
grating (0 = 0), and increases on both sides of this point. It is prac- 
tically constant, however, for a considerable region near the normal, 
because here the cosine is varying slowly. A common value for Ft is 21 ft, 
and a concave grating with this radius of curvature is called a 21-// grating. 
Two other common mountings for concave gratings are the Rowland 
mounting and the Eagle mounting, illustrated in Fig. 17 A'. In the Row- 
land mounting, which is now mostly of historical interest, the grating G 



THE DIFFRACTION GRATING 



349 



and plate holder P are fixed to opposite ends of a rigid beam of length R. 
The two ends of this beam rest on swivel trucks which are free to move 
along two tracks at right angles to each other. The slit S is mounted 
just above the intersection of the two tracks. With this arrangement, 
the portion of the spectrum reaching the plate may be varied by sliding 
the beam one way or the other, thus varying the angle of incidence i. 
It will be seen that this effectively moves *S around on the Rowland circle. 
For any setting the spectrum will be in focus on P, and it will be nearly a 
normal spectrum (Sec. 17.6) because the angle of diffraction fl~0. The 
track SP is usually graduated in wavelengths since, as may be easily 



Rowland mounting 




Eagle mounting 



L 



W^WMmmmSZ^ 



*k 



/ 



\ S = Slit 
G - Grating 
P = Plate 




(a) (c) 

Fig. 17 N. (a) One of the earliest and (6) one of the commonest forms of concave- 
grating spectrograph, (c) Mounting for plane reflection grating. 



shown from the grating equation, the wavelength in a given order arriv- 
ing at P is proportional to the distance SP. 

The Eagle mounting, because of its compactness and flexibility, has 
largely replaced the Rowland and Paschen forms. Here the part of the 
spectrum is observed which is diffracted back at angles nearly equal to the 
angle of incidence. The slit S is placed at one end of the plateholder, 
the latter being pivoted like a gate at S. To observe different portions 
of the spectrum, the grating is turned about an axis perpendicular to the 
figure. It must then be moved along horizontal ways, and the plate- 
holder turned, until P and S again lie on the Rowland circle. The 
instrument can be mounted in a long box or room where the temperature 
is held constant. Variations of temperature displace the spectrum lines 
owing to the change of grating space which results from the expansion or 
contraction of the grating. With a grating of speculum metal it can be 
shown that a change of temperature of 0.1°C shifts a line of wavelength 



350 PHYSICAL OPTICS 

5000 A in any order by 0.013 A.. The Eagle mounting is commonly used 
in vacuum spectrographs for the investigation of ultraviolet spectra in the 
region below 2000 A. Since air absorbs these wavelengths, the air must 
be pumped out of the spectrograph, and this compact mounting is con- 
venient for the purpose. The Paschen mounting is also frequently used 
in vacuum spectrographs with the light incident on the grating at a 
practically grazing angle. The Littrow mounting, also shown in Fig. 
17 N, is the only common method of mounting large plane reflection 
gratings. In principle it is very much like the Eagle mounting, the 
main difference being that a large achromatic lens renders the incident 
light parallel and focuses the diffracted light on P, so that it acts as both 
collimator and telescope lenses at once. 

One important drawback of the concave grating as used in the mount- 
ings described above is the presence of strong astigmatism. It is least 
in the Eagle mounting. This defect of the image always occurs when 
a concave mirror is used off axis. Here it has the consequence that each 
point on the slit is imaged as two lines, one located on the Rowland circle 
perpendicular to its plane, the other in this plane and at some distance 
behind the circle. If the slit is accurately perpendicular to the plane, 
the sharpness of the spectrum lines is not seriously impaired by astig- 
matism. Because of the increased length of the lines, however, some 
loss of intensity is involved. More serious is the fact that it is impossible 
to study the spectrum of different parts of a source, or to separate Fabry- 
Perot rings, by projecting an image on the slit of the spectrograph. For 
this purpose, a stigmatic mounting is required. The commonest of these 
is the Wadsworth mounting, in which the concave grating is illuminated 
by parallel light. The light from the slit is rendered parallel by a large 
concave mirror, and the spectrum is focused at a distance of about one- 
half the radius of curvature of the grating. 

PROBLEMS 

1. Derive Eq. 17c, as suggested in Sec. 17.2, by integrating Eq. 156 over the proper 

limits. 

2. An ideal transmission grating has d = 36. Described the condition of the vibra- 
tion curve at a point corresponding to the first missing order. 

Ans. Curve for each slit makes a closed circle. 

3. Make qualitative sketches of the intensity patterns for (a) four slits having 
d/b = 7, and (6) nine slits having d/b = 3. Label several points on the axis of 
abscissas with the corresponding values of /3 and 7. 

4. Prove that the intensity formula for the ideal grating reduces to that for the 
double slit in the special case N = 2. (Mint: Apply the trigonometric formula for the 
sine of the double angle.) 

6. Seven sources of microwaves (X = 3 cm) are placed side by side, 8 cm apart. 
Describe the radiation pattern observed at a distance sufficient to ensure Fraunhofer 



THE DIFFRACTION GRATING 351 

diffraction. Compute the angular half width of the central maximum. Find also 
the angular separation of the principal maxima and of the subsidiary maxima. 

6. Prove that the intensity pattern for N slits can be represented as the sum over 
all orders of a number of single-slit patterns of the type that would be produced by 
an aperture of width Nd. (The general proof, though exact, is difficult. Try sum- 
ming the numerical values of the secondary maxima for a specific case, say, N = 4, 
and compare with the values calculated from the grating formula.) 

7. Let light of two wavelengths, 5200 and 5500 A, fall on a plane transmission grat- 
ing having 3500 lines per centimeter. The emerging parallel light is to be focused 
on a screen by a lens of 1.5 m focal length. Find the distance on the screen in centi- 
meters between the two spectrum lines (a) in the first order, (b) in the third order. 

8. Find the minimum number of lines that a diffraction grating would need to have 
in order to resolve in the first order the red doublet given by a mixture of hydrogen 
and deuterium. The wavelength difference is 1.8 A at X6563. Ans. 3,647. 

9. Compare, with regard to chromatic resolving power and angular dispersion 
(a) a diffraction grating ruled with a total of 40,000 lines in a distance of 5 cm, when 
used in the first order at X6250, and (6) a glass prism 5 cm on each side, the glass having 
n = 1.5900 at X6000 and n = 1.5880 at X6500. 

10. Calculate the angular dispersion in degrees per angstrom for a diffraction grat- 
ing having 14,438 lines per inch, when used in the third order at 4200 A. Assume 
normal incidence. Ans. 0.014°/A. 

11. Describe the characteristics that would be desired for a filter to remove the 
other orders that overlap the region X3000 in the third order of a grating spectrum. 

12. It is desired to study the structure of a band in the neighborhood of 4300 A, 
using a 6-in. plane grating having 30,000 lines per inch, and mounted in the Littrow 
system. Find (a) the highest order that can be used, (6) the angle of incidence 
required to observe it, (c) the smallest wavelength interval resolved, and (d) the plate 
factor, if the lens has a focal length of 3 m. 

Ans. (a) m = 3. (6) 49°37^'. (c) 0.008 A. (d) 0.609 A/mm. 

13. A transmission grating having d = 1.65 X 10~ 4 cm is illuminated at various 
angles of incidence by light of wavelength 6000 A. Make a plot of the deviation of 
the first-order diffracted beam from the direction of the incident light, using the angle 
of incidence from to 90° as abscissas. 

14. What would be the order number and resolving power for a reflection echelon 
having 30 plates each 12 mm thick, if it were illuminated by fight of the mercury 
resonance fine, X2537? Ans. 94,600. 2.84 X 10". 

15. An echelette grating has 1200 lines per inch, and is ruled for concentration at a 
wavelength of 6 p in the first order, (a) Find the angle of the ruled faces to the plane 
of the grating, (b) Find the angular dispersion at this wavelength, assuming normal 
incidence, (c) If this grating were illuminated by the green mercury line, what 
order or orders would be observed? 

16. Prove that one can express the resolving power of an echelle grating as X/AX = 
(2#/X)[r 2 /(l + r 8 )]*, where B is the width of the grating, and r = t/b the ratio of the 
depth of the steps to their width. It is assumed that the light is incident and dif- 
fracted normal to the faces of width b. (Hint: Use the principle that the resolving 
power equals the number of wavelengths in the path difference between the rays 
from opposite edges of the grating.) 

17. Investigate the deviation from linear dispersion in the case of a concave grating 
of 15 ft radius used in the Rowland mounting. If the photographic plate is 18 in. long, 
by what per cent does the dispersion at one end differ from that, at the center? What 
error in angstroms would be made by computing a wavelength at the end of the plalte 



352 PHYSICAL OPTICS 

by using the dispersion at the center? Assume that X3660 in the first order occurs 
at the center and that the grating has 15,000 lines per in. 

18. A concave grating of 21 ft radius is incorporated in an Eagle mounting. The 
grating has 15,000 lines per inch, and is 5^ in. wide. If the angle of incidence is 37°, 
find what wavelength in the second order falls next to the slit. Compute the resolving 
power and plate factor, also in the second order, at a point on the plate that is 20 cm 
from the slit along the Rowland circle, in the direction of the grating normal. 

Ans. 10,191 A. 157,500. 1.12 A/mm. 



CHAPTER 18 

FRESNEL DIFFRACTION 



The diffraction effects obtained when either the source of light or the 
observing screen, or both, are at a finite distance from the diffracting 
aperture or obstacle come under the classification of Fresnel diffraction- 
These effects are the simplest to observe experimentally, the only appara- 
tus required being a small source of light, the diffracting obstacle, and 
a screen for observation. In the Fraunhofer effects discussed in the 
preceding chapters, lenses were required to render the light parallel, and 
to focus it on the screen. Now, however, we are dealing with the more 
general case of divergent light which is not altered by any lenses. Since 
Fresnel diffraction is the easiest to observe, it was historically the first 
type to be investigated, although its explanation requires much more 
difficult mathematical theory than that necessary in treating the plane 
waves of Fraunhofer diffraction. In this chapter we consider only some 
of the simpler cases of Fresnel diffraction, which are amenable to explana- 
tion by fairly direct mathematical and graphical methods. 

18.1. Shadows. One of the greatest difficulties in the early develop- 
ment of the wave theory of light lay in the explanation of the observed 
fact that light appears to travel in straight lines. Thus if we place an 
opaque object in the path of the light from a point source, it casts a 
shadow having a fairly sharp outline of the same shape as the object. 
It is true, however, that the edge of this shadow is not absolutely sharp 
and that when examined closely it shows a system of dark and light bands 
in the immediate neighborhood of the edge. In the days of the corpuscu- 
lar theory of light, attempts were made by Grimaldi and Newton to 
account for such small effects as due to the deflection of the light cor- 
puscles in passing close to the edge of the obstacle. The correct explana- 
tion in terms of the wave theory we owe to the brilliant work of Fresnel. 
In 1815 he showed not only that the approximately rectilinear propaga- 
tion of light could be interpreted on the assumption that light is a wave 
motion, but also that in this way the diffraction fringes could in many 
cases be accounted for in detail. 

To bring out the difficulty encountered in explaining shadows by the 
wave picture, let us consider first the passage of divergent light through 

353 



354 



PHYSICAL OPTICS 



an opening in a screen. In Fig. 18-4 the light originates from a small 
pinhole H, and a certain portion M N of the divergent wave front is 
allowed to pass the opening. According to Huygens' principle, we may 
regard each point on the wave front as a source of secondary wavelets. 
The envelope of these at a later instant gives a divergent wave with H 
as its center and included between the lines HE and HF. This wave as 
it advances will produce strong illumination in the region EF of the screen. 
But also part of each wavelet will travel into the space behind LM and 
NO, and hence might be expected to produce some light in the regions 



* 



// 




// 



B \P 




Fig. 18A. Huygens' principle applied to 
secondary wavelets from a narrow 
opening. 



* 



Fig. 18B. The obliquity factor for Huy- 
gens' secondary wavelets. 



of the geometrical shadow outside of E and F. Common experience 
shows that there is actually no illumination on these parts of the screen, 
except in the immediate vicinity of E and F. According to Fresnel, 
this is to be explained by the fact that in the regions well beyond the 
limits of the geometrical shadow the secondary wavelets arrive with 
ohase relations such that they interfere destructively and produce prac- 
tically complete darkness. 

The secondary wavelets cannot have uniform amplitude in all direc- 
tions, since if this were so, they would produce an equally strong wave 
in the backward direction. In Fig. 18A the envelope on the left side 
of the screen would represent a reverse wave converging toward H. 
Obviously such a wave does not exist physically, and hence one must 
assume that the amplitude at the back of a secondary wave is zero. 
The more exact formulation of Huygens' principle to be mentioned later 
(Sec. 18.17) justifies this assumption, and also gives quantitatively the 
variation of the amplitude with direction. The so-called obliquity factor, 
as is illustrated in Fig. \SB, requires an amplitude varying as 1 + cos 6, 



FRESNEL DIFFRACTION 



355 



where 6 is the angle with the forward direction. At right angles, in the 
directions P and Q of the figure, the amplitude falls to one-half, and the 
intensity to one-quarter, of its maximum value. Another property that 
the wavelets must be assumed to have, in order to give the correct 
results, is an advance of phase of one-quarter period ahead of the wave 
that produces them. The consequences of these two rather unexpected 
properties, and the manner in which they are derived, will be discussed 
later. 

18.2. Fresnel's Half-period Zones. As an example of Fresnel's 
approach to diffraction problems, we first consider his method of finding 
the effect that a slightly divergent spherical wave will produce at a point 




D 

Fig. 18C. Construction of half-period 
zones on a spherical wave front. 



Fig. 18£>. Path difference A at a distance 
s from the pole of a spherical wave. 



ahead of the wave. In Fig. 18C let BCDE represent a spherical wave 
front of monochromatic light traveling toward the right. Every point 
on this sphere may be thought of as the origin of secondary wavelets, 
and we wish to find the resultant effect of these at a point P. To do 
this, we divide the wave front into zones by the following construction: 
Around the point 0, which is the foot of the perpendicular from P, we 
describe a series of circles whose distances from 0, measured along the 
arc, are Si, S2, s 3 , . . . , s m and are such that each circle is a half wave- 
length farther from P. If the distance OP = b, the circles will be at 
distances b + A/2, b + 2X/2, b + 3X/2, . . . , b + mX/2 from P. 

The areas S m of the zones, i.e., of the rings between successive circles, 
are practically equal. In proving this, we refer to Fig. 18 D, where a sec- 
tion of the wave spreading out from H is shown with radius a. If a 
circle of radius b is now drawn (broken circle) with its center at P and 
tangent to the wave front at its "pole" 0, the path HQP is longer than 
HOP by the segment indicated by A. For the borders of the zones, this 
path difference must be a whole multiple of X/2. To evaluate it, we note 
first that in all optical problems the distance s is small compared with 
a and b. Then s may be considered as the vertical distance of Q above 



356 » PHYSICAL OPTICS 

the axis, and A may be equated to the sum of the sagittas of the two arcs 
OQ and OR. By the sagitta formula we then have 

The radii s m of the Fresnel zones are such that 

^ 2 a + ^ ,,£,,« 

OT 2 = S " W < m > 

and the area of any one zone becomes 

&. = *,« - 8 _,») = , I ( a -Mj) - ^ rtX (18c) 

To the approximation considered, it is therefore constant and independent 
of m. A more exact evaluation would show that the area increases very 
slowly with m. 

By Huygens' principle we now regard every point on the wave as 
sending out secondary wavelets in the same phase. These will reach P 
with different phases, since each travels a different distance. The phases 
of the wavelets from a given zone will not differ by more than t, and 
since each zone is on the average X/2 farther from P, it is clear that the 
successive zones will produce resultants at P which differ by ir. This 
statement will be examined in more detail in Sec. 18.6. The difference 
of half a period in the vibrations from successive zones is the origin of 
the name half-period zones. If we represent by A m the resultant ampli- 
tude of the light from the with zone, the successive values of A m will have 
alternating signs, because changing the phase by ir means reversing the 
direction of the amplitude vector. Calling the resultant amplitude due 
to the whole wave A, it may be then written as the sum of the series 

A = A x - At + Ai - A 4 + • * • + (-l)"'- l A m (I8d) 

There are three factors which determine the magnitudes of the suc- 
cessive terms in this series: First, because the area of each zone determines 
the number of wavelets it contributes, the terms should be approximately 
equal but should increase slowly; second, since the amplitude decreases 
inversely with the average distance from P of the zone, the magnitudes 
of the terms are reduced by an amount which increases with m; and 
third, because of the increasing obliquity, their magnitudes should 
decrease. Thus we may express the amplitude due to the rath zone as 

A m = const. • j> (1 + cos 0) (18e) 

where d m is the average distance to P and the angle at which the light 



FRESNEL DIFFRACTION 357 

leaves the zone. It appears in the form shown because of the obliquity 
factor assumed in the preceding section. Now an exact calculation of 
the <S m 's shows that the factor b in Eq. 18c must be replaced by b + A, 
where A is the path difference for the middle of the zone. Since at the 
same time d m = b + A, we find that the ratio S m /d m is a constant, inde- 
pendent of m. Therefore we have left only the effect of the obliquity 
factor 1 + cos 8, which causes the successive terms in Eq. lSd to decrease 
very slowly. The decrease is least slow at first, because of the rapid 
change of with m, but the amplitudes soon become nearly equal. 

With this knowledge of the variation in magnitude of the terms, we 
may evaluate the sum of the series by grouping its terms in the following 
two ways. Supposing m to be odd, 






An 

2 



-^+A m (18/) 

Now since the amplitudes A x , A 2 , . . . do not decrease at a uniform rate, 
each one is smaller than the arithmetic mean of the preceding and follow- 
ing ones. Therefore the quantities in parentheses in the above equations 
are all positive, and the following inequalities must hold: 

Because of the fact that the amplitudes for any two adjacent zones are 
very nearly equal, it is then possible to equate A x to A 2 , and A m -\ to A m . 
The result is 

A - T + 4f im) 

If m is taken to be even, we find by the same method that 

A\ A m _ . 
T~T " A 

Hence the conclusion is that the resultant amplitude at P due to m 
zones is either half the sum or half the difference of the amplitudes con- 
tributed by the first and last zones. If we allow m to become large enough 
so that the entire spherical wave is divided into zones, 6 approaches 
180° for the last zone. Therefore the obliquity factor causes A m to 
become negligible, and the amplitude due to the whole wave is iust half 
that due to the first zone acting alone. 



358 



PHYSICAL OPTICS 



Figure 18E shows how these results may be understood from a graphical 
construction. The vector addition of the amplitudes .1 h At, A 3 , . . . , 
which are alternately positive and negative, would normally be performed 
by drawing them along the same line, but here for clarity they are sep- 
arated in a horizontal direction. The tail of each vector is put at the 
same height as the head of the previous one. Then the resultant ampli- 
tude A due to any given number 






4m- 



MA 



Pi- o o 



Fig. 18/?. Addition of the amplitudes 
from half-period zones. 



of zones will be the height of the 
final arrowhead above the horizon- 
tal base line. In the figure, it is 
.di shown for 12 zones, and also for a 
very large number of zones. 

18.3. Diffraction by a Circular 
Aperture. Let us examine the 
effect upon the intensity at P (Fig. 



Diffracting 
screen 



18C) of blocking off the wave by a screen pierced by a small circular aper- 
ture as shown in Fig. 18F. If the radius of the hole r = OR is made 
equal to the distance s x to the outer edge of the first half-period zone,* 
the amplitude will be A { and this is twice the amplitude due to the 
unscreened wave. Thus the intensity at P is four times as great as if the 
screen were absent. Increasing 
the radius of the hole until it in- 
cludes the first two zones, the 
amplitude is A x — A 2 , or practi- 
cally zero. The intensity has 
actually fallen to almost zero by 
increasing the size of the hole. A 
further increase of r will cause the 
intensity to pass through maxima 
and minima each time the number 
of zones included becomes odd or 
even. 

The same effect is produced by 
moving the point of observation P 

continuously toward or away from the aperture along the perpendicular. 
This varies the size of the zones, so that if P is originally at a position 
such that PR — PO of Fig. 18/'' is X/2 (one zone included), moving P 
toward the screen will increase this path difference to 2X/2 (two zones), 
3X/2 (three zones), etc. We thus have maxima and minima along the 
axis of the aperture. 

* We are here assuming that the radius of curvature of the wave striking the screen 
is large, so that distances measured along the chord may be taken as equal to those 
measured along the arc. 




Fig. 18F. Geometry for 
through a circular opening. 



light passing 



FRESNEL DIFFRACTION 



359 



The above considerations give no information about the intensity at 
points off the axis. A mathematical investigation, which we shall not 
discuss because of its complexity,* shows that P is surrounded by a system 
of circular diffraction fringes. Several photographs of these fringes are 
illustrated in Fig. 18G. These were taken by placing a photographic 
plate some distance behind circular holes of various sizes, illuminated by 
monochromatic light from a distant point source. Starting at the 
upper left of the figures, the holes were of such sizes as to expose one, 
two, three, etc., zones. The alternation of the center of the pattern from 




Fig. 18G. Diffraction of light by small circular openings. {Original photographs by 
Hufford.) 

bright to dark illustrates the result obtained above. The large pattern 
on the right was produced by an aperture containing 71 zones. 

18.4. Diffraction by a Circular Obstacle. When the hole is replaced 
by a circular disk, Fresnel's method leads to the surprising conclusion 
that there should be a bright spot in the center of the shadow. For a 
treatment of this case, it is convenient to start constructing the zones at 
the edge of the disk. If, in Fig. ISF, PR = d, the outer edge of the first 
zone will be d -+- (A/2) from P, of the second d + (2X/2), etc. The sum 
of the series representing the amplitudes from all the zones in this case is, 
as before, half the amplitude from the first exposed zone. In Fig. 18/? 
it would be obtained by merely omitting the first few vectors. Hence the 
intensity at P is practically equal to that produced by the unobstructed 
wave. This holds only for a point on the axis, however, and off the axis 
the intensity is small, showing faint concentric rings. In Fig. lSH(a) and 
(b), which shows photographs of the bright spot, these rings are unduly 
strengthened relative to the spot by overexposure. In (c) the source, 
instead of being a point, was a photographic negative of a portrait of 

* See T. Preston, "Theory of Light," 5th ed., pp. 324-327, The Macmillan Com- 
pany, New York, 1928. 



360 



PHYSICAL OPTICS 



Woodrow Wilson on a transparent plate, illuminated from behind. The 
disk acts like a rather crude lens in forming an image, since for every point 
in the object there is a corresponding bright spot in the image. 

The complete investigation of diffraction by a circular obstacle shows 
that, besides the spot and faint rings in the shadow, there are bright cir- 
cular fringes bordering the outside of the shadow. These arc similar in 




6 




_ (a) (b) 

Fig. 1SH. Diffraction by a circular obstacle, (a) and (6) Point source, (c) A nega- 
tive of Woodrow Wilson as a source. (After Hufford.) 

origin to the diffraction fringes from a straight edge to be investigated in 
Sec. 18.11. 

The bright spot in the center of the shadow of a 1-cent piece may 
be seen by examining the region of the shadow produced by an arc 
light several meters away, preferably using a magnifying glass. The 
spot is very tiny in this case, and difficult to find. It is easier to see 
with a smaller object, such as a ball bearing. 

18.6. Zone Plate. This is a special screen designed to block off the 
light from every other half-period zone. The result is to remove either 

all the positive terms in Eq. 18d or 
all the negative terms. In either 
case the amplitude at P (Fig. ISC) 
will be increased to many times its 
value in the above cases. A zone 
plate can easily be made in practice 
by drawing concentric circles on 
white paper, with radii proportional 
to the square roots of whole num- 
bers (see Fig. 187). Every other zone is then blackened, and the result 
is photographed on a reduced scale. The negative, when held in the light 
from a distant point source, produces a large intensity at a point on 
its axis at a distance corresponding to the size of the zones and the wave- 
length of the light used. The relation between these quantities is con- 
tained in Eq. ISb, which for the present purpose may be written 




Fig. 18/. Zone plates. 



X sjfl . 1 

m 2 = ^\a + b 



(18ft) 



Hence we see that, for given a, b, and X, the zones must have s m ^^ s/m. 



FRESNEL DIFFRACTION 361 

The bright spot produced by a zone plate is so intense that the plate 
acts much like a lens. Thus suppose that the first 10 odd zones are 
exposed, as in the zone plate of Fig. 187(a). This leaves the amplitudes 
A if Az, At, . . . , A ig (see Fig. 18E), the sum of which is nearly ten times 
A i. The whole wave front gives ?Ai, so that, using only 10 exposed 
zones, we obtain an amplitude at P which is 20 times as great as when the 
plate is removed. The intensity is therefore 400 times as great. If the 
odd zones are covered, the amplitudes A 2, A 4, A 6 , . . . will give the 
same effect. The object and image distances obey the ordinary lens 
formula, since, by Eq. 18h, 

1 1 = mX = 1 
a b s m 2 f 

the focal length / being the value of 6 for a = 00 1 namely, 

There are also fainter images corresponding to focal lengths //3, f/5, f/7, 
. . . , because at these distances each zone of the plate includes 3, 5, 7, 
. . . Fresnel zones. When it includes three, for example, the effects 
of two of them cancel but that of the third is left over. 

18.6. Vibration Curve for Circular Division of the Wave Front. Our 
consideration of the vibration curve in the Fraunhofer diffraction by a 
single slit (Sec. 15.4) was based upon the division of the plane wave front 
into infinitesimal elements of area which were actually strips of infinitesi- 
mal width parallel to the length of the diffracting slit. The vectors 
representing the contributions to the amplitude from these elements were 
found to give an arc of a circle. This so-called strip division of the wave 
front is appropriate when the source of light is a narrow slit and the 
diffracting aperture rectangular. The strip division of a divergent wave 
front from such a source will be discussed below (Sec. 18.8). The method 
of dividing the spherical wave from a point source appropriate to any case 
of diffraction by circular apertures or obstacles involves infinitesimal 
circular zones. 

Let us consider first the amplitude diagram when the first half-period 
zone is divided into eight subzones, each constructed in a manner similar 
to that used for the half-period zones themselves. We make these sub- 
zones by drawing circles on the wave front (Fig. 18C) which are distant 

7,-L 1X >, _L 2X /, . 3X ». , x 

6 + 82' 6+ 82 ,6 + 82' • ■ • ' 6 + 2 

from P. The light arriving at P from various points in the first subzone 
will not vary in phase by more than x/8. The resultant of these may be 
represented by the vector Oi in Fig. 18.K(a). To this is now added a 2t 



362 



PHYSICAL OPTICS 



the resultant amphauae due to the second subzone, then a 3 due to the 
third subzone, etc. The magnitudes of these vectors will decrease very 
slowly as a result of the obliquity factor. The phase difference 6 between 
each successive one will be constant and equal to w/8. Addition of all 
eight subzones yields the vector AB as the resultant amplitude from the 
first half-period zone. Continuing this process of subzoning to the 
second half-period zone, we find CD as the resultant for this zone, and 
A D as that for the sum of the first two zones. These vectors correspond 
to those of Fig. 18E. Succeeding half-period zones give the rest of the 
figure, as shown. 




A 

(a) (6) 

Fig. 18/. Vibration spiral for Fresnel half -period zones of a circular opening. 

The transition to the vibration curve of Fig. 18./ (6) results from 
increasing indefinitely the number of subzones in a given half-period 
zone. The curve is now a vibration spiral, eventually approaching Z 
when the half-period zones cover the whole SDherical wave. Any one 
turn is very nearly a circle, but does not quite close because of the slow 
decrease in the magnitudes of the individual amplitudes. The sig- 
nificance of the series of decreasing amplitudes, alternating in sign, used 
in Sec. 18.2 for the half-period zones, becomes clearer when we keep in 
mind the curve of Fig. 18J(6). It has the additional advantage of allow- 
ing us to determine directly the resultant amplitude due to any fractional 
number of zones. It should be mentioned in passing that the resultant 
amplitude AZ, which is just half the amplitude due to the first half-period 
zone, turns out to be, from this treatment, 90° in phase behind the light 
from the center of the zone system. This cannot be true, since it is 
impossible to alter the resultant phase of a wave merely by the artifice 
of dividing it into zones and then recombining the effects of these. The 
discrepancy is a defect of Fresnel's theory resulting from the approxima- 



FRESNEL DIFFRACTION 



3o3 



tions made therein, and does not occur in the more rigorous mathematical 
treatment (see Sec. 18.17). 

18.7. Apertures and Obstacles with Straight Edges. If the configura- 
tion of the diffracting screen, instead of having circular symmetry, 
involves straight edges like those of a slit or wire, it is possible to use 
as a source a slit rather than a point. The slit is set parallel to these 
edges, so that the straight diffraction fringes produced by each element 
of its length are all lined up on the observing screen. A considerable 
gain of intensity is achieved thereby. In the investigation of such cases, 
it is possible to regard the wave front as cylindrical, as shown in Fig. 1SK. 




Fig. 1SK. Cylindrical wave from a slit which is illuminated coherently. Half-period 
strips are marked on the wave front. 

It is true that to produce such a cylindrical envelope to the Huygens 
wavelets emitted by various points on the slit these must emit coherently, 
and in practice this will not usually be true. Nevertheless, when intensi- 
ties are added, as is required for noncoherent emission, the resulting 
pattern is the same as would be produced by a coherent cylindrical wave. 
In the following treatment of problems involving straight edges, we shall 
therefore make the simplification of assuming the source slit to be 
illuminated by a parallel monochromatic beam, so that it emits a truly 
cylindrical wave. 

18.8. Strip Division of the Wave Front. The appropriate method of 
constructing half-period elements on a cylindrical wave front consists in 
dividing the latter into strips, the edges of which are successively one-half 
wavelength farther from the point P (Fig. ISK). Thus the points M , 
Mi, M2, ... on the circular section of the cylindrical wave are at dis- 
tances b, b + (X/2), b + (2X/2), . . . from P. M is on the straight 
line SP. The half-period strips MoM lt M\M? ; . . . now stretch along 
the wave front parallel to the slit. We may call this procedure strip 
division of the wave front. 



364 



PHYSICAL OPTICS 



In the Fresnel zones obtained by circular division, the areas of the 
zones were very nearly equal. With the present type of division this 
is by no means true. The areas of the half-period strips are proportional 
to their widths, and these decrease rapidly as we go out along the wave 
front from M . Since this effect is much more pronounced than any 
variation of the obliquity factor, the latter need not be considered. 

The amplitude diagram of Fig. 18L(a) is obtained by dividing the 
strips into substrips in a manner analogous to that described in Sec. 18.6 
for circular zones. Dividing the first strip above M into nine parts, 





(a) (b) 

Fig. 18L. Amplitude diagrams for the formation of Cornu's jpiral. 

we find that the nine amplitude vectors from the substrips extend from 
to B, giving a resultant A y = OB, for the first half-period strip. The 
second half-period strip similarly gives those between B and C, with a 
resultant A 2 = BC. Since the amplitudes now decrease rapidly, A 2 is 
considerably smaller than A i, and their difference in phase is appreciably 
greater than tt. A repetition of this process of subdivision for the succeed- 
ing strips on the upper half of the wave gives the more complete diagram 
of Fig. 18L(6). Here the vectors are spiraling in toward Z, so that the 
resultant for all half-period strips above the pole M becomes OZ. 

18.9. Vibration Curve for Strip Division. Cornu's Spiral. When we 
go over to elementary strips of infinitesimal width, we obtain the vibra- 
tion curve as a smooth spiral, part of which is shown in Fig. ISM. The 
complete curve representing the whole wave front would be carried 
through many more turns, ending at the points Z and Z' . Only the 
part from to Z was considered above. The lower half, Z'O, arises from 
the contributions from the half-period strips below M . 

This curve, called Cornu's* spiral, is characterized by the fact that the 

* A. Cornu (1841-1902). Professor of experimental physics at the £cole Poly- 
technique, Paris. 



FRESNEL DIFFRACTION 



365 



angle 8 it makes with the x axis is proportional to the square of the dis- 
tance v along the curve from the origin. Remembering that, in a vibra- 
tion curve, 5 represents the phase lag in the light from any element 




Fig. 18M. Cornu's spiral, drawn to include five half-period zones on either side of the 
pole. 

of the wave front, we obtain this definition of the curve by using Eq. 18a 
for the path difference, as follows: 



2tt ?r(a + b) ir . 

X a&X 2 



(18/) 



Here we have introduced a new variable for use in plotting Cornu's 
spiral, namely, 



f 2(q + 6) 

V = S \l--aW- 



(18fc) 



It is defined in such a way as to make it dimensionless, so that the same 
curve may be used for any problem, regardless of the particular values of 
a, b, and X. 

18.10. Fresnel's Integrals. The x and y coordinates of Cornu's spiral 
may be expressed quantitatively by two integrals, and a knowledge of 



366 PHYSICAL OPTICS 

these will permit accurate plotting and calculations. They are derived 
most simply as follows : Since the phase difference d is the angle determin- 
ing the slope of the curve at any point (see Fig. 18M), the changes in the 
coordinates for a given small displacement dv along the spiral are given by 

i . irv 2 j 

ax = dv cos o = cos -=- dv 



dy = dv sin 5 = sin -%- dv 



2 

-r- 
2 



where the value of 8 from Eq. 18; has been introduced. Thus the coordi- 
nates of any point (x,y) on Cornu's spiral become 

f" TV 2 

x = I cos-s- dv (181) 

Jo * 

y = Psin^dv (18m) 

Jo & 

These are known as Fresnel's integrals. They cannot be integrated in 
closed form, but yield infinite series which may be evaluated in several 
ways.* Although the actual evaluation is too complicated to be given 
here, Ave have included a table of the numerical values of the integrals 
(Table 18-1). Later on, in Sec. 18.14, the method of using these in 
accurate computations of diffraction patterns is explained. 

Let us first examine some features of the quantitative Cornu's spiral 
of Fig. 18iV, which is a plot of the two Fresnel integrals. The coordinates 
of any point on the curve give their values for a particular upper limit v 
in Eqs. 18Z and 18m. The scale of v is marked directly on the curve, 
and has equal divisions along its length. Particularly useful to remember 
are the positions of the points v = \, V%, and 2 on the curve. They 
represent one-half, one, and two half-period strips, respectively, as may 
be verified by computing the corresponding values of 5 from Eq. I8j. 
More important, however, are the coordinates of the end points Z' and Z. 
They are (— it - *) and (^,-£), respectively. 

As with any vibration curve, the amplitude due to any given portion 
of the wave front may be obtained by finding the length of the chord 
of the appropriate segment of the curve. The square of this length 
then gives the intensity. Thus the Cornu's spiral of Fig. 18.V may be 
used for the graphical solution of diffraction problems, as will be illus- 
trated below. It is to be noted at the start, however, that the numerical 
values of intensities computed in this way are relative to a value of 2 
for the unobstructed wave. Thus, if A represents any amplitude obtained 

* For the methods of evaluating Fresnel's integrals, see R. W. Wood, "Physical 
Optics," 2d ed., p. 247, The Macmillau Company, New York, 1921. 



FRESNEL DIFFRACTION 
Table 18-1. Table of Fresnel Integrals 



367 



V 


X 


y 


V 


X 


y 


0.00 


0.0000 


0.0000 


4.50 


0.5261 


0.4342 


0.10 


0.1000 


0.0005 


4.60 


0.5673 


0.5162 


0.20 


0.1999 


0.0042 


4.70 


0.4914 


0.5672 


0.30 


0.2994 


0.0141 


4.80 


0.4338 


0.4968 


0.40 


0.3975 


0.0334 


4.90 


0.5002 


0.4350 


0.50 


0.4923 


0.0647 


5.00 


0.5637 


0.4992 


0.60 


0.5811 


0.1105 


5.05 


0.5450 


0.5442 


0.70 


0.6597 


0.1721 


5.10 


0.4998 


0.5624 


0.80 


0.7230 


0.2493 


5.15 


0.4553 


0.5427 


0.90 


0.7648 


0.3398 


5.20 


0.4389 


0.4969 


1.00 


0.7799 


0.4383 


5.25 


0.4610 


0.4536 


1.10 


0.7638 


0.5365 


5.30 


0.5078 


0.4405 


1.20 


0.7154 


0.0234 


5.35 


0.5490 


0.4662 


1.30 


0.6386 


0.6863 


5.40 


0.5573 


0.5140 


1.40 


0.5431 


0.7135 


5.45 


0.5269 


0.5519 


1.50 


0.4453 


0.6975 


5.50 


0.4784 


0.5537 


1.60 


0.3655 


0.6389 


5.55 


i 0.4456 


0.5181 


1.70 


0.3238 


0.5492 


5.60 


0.4517 


0.4700 


1.80 


0.3336 


0.4508 


5.65 


0.4926 


0.4441 


1.90 


0.3944 


0.3734 


5.70 


0.5385 


0.4595 


2.00 


0.4882 


0.3434 


5.75 


0.5551' 


. 5049 


2.10 


0.5815 


0.3743 


5.80 


0.5298 


0.5461 


2.20 


0.6363 


0.4557 


5.85 


0.4819 


0.5513 


2.30 


0.6266 


0.5531 


5.90 


0.4486 


0.5163 


2.40 


0.5550 


0.6197 


5.95 


0.4566 


0.4688 


2.50 


0.4574 


0.6192 


6.00 


0.4995 


0.4470 


2.60 


0.3890 


0.5500 


G.05 


0.5424 


0.4689 


2.70 


0.3925 


0.4529 


6.10 


0.5495 


0.5165 


2.80 


0.4675 


0.3915 


6.15 


0.5146 


0.5496 


2.90 


0.5624 


0.4101 


6.20 


. 4676 


0.5398 


3.00 


0.6058 


0.4963 


6.25 


0.4493 


0.4954 


3.10 


0.5616 


0.5818 


6.30 


0.4760 


. 4555 


3.20 


0.4664 


0.5933 


6.35 


0.5240 


0.4560 


3.30 


0.4058 


0.5192 


6.40 


0.5496 


0.4965 


3.40 


0.4385 


0.4296 


6.45 


0.5292 


0.5398 


3.50 


0.5326 


0.4152 


6.50 


0.4816 


0.5454 


3.60 


0.5880 


0.4923 


6.55 


0.4520 


0.5078 


3.70 


0.5420 


0.5750 


6.60 


0.4690 


0.4631 


3.80 


0.4481 


0.5656 


6.65 


0.5161 


0.4549 


3.90 


0.4223 


0.4752 


6.70 


0.5467 


0.4915 


4.00 


0.4984 


0.4204 


6.75 


0.5302 


0.5362 


4.10 


0.5738 


0.4758 


6.80 


0.4831 


. 5436 


4.20 


0.5418 


0.5633 


6.85 


0.4539 


0.5060 


4.30 


0.4494 


0.5540 


6.90 


0.4732 


0.4624 


4.40 


0.4383 


0.4622 


6.95 


0.5207 


0.4591 



368 



PHYSICAL OPTICS 



from the plot, the intensity /, expressed as a fraction of that which would 
exist were no screen present, which we shall call I , is 



— = iA 2 



(18n) 



To verify this statement, we note that according to the discussion of Sec. 
18.8 a vector drawn from to Z gives the amplitude due to the upper 




Fig. 18.V. Cornu's spiral, a plot of the Fresnel integrals. 

half of the wave. Similarly, one from Z' to gives that due to the lower 
half. Each of these has a magnitude l/v / 2, so that when they are added, 
and the sum is squared to obtain the intensity due to the whole wave, 
we find that I a = 2, with the conventional scale of coordinates used in 
Fig. 1SN.* 

* It will be noticed that the phase of the resultant wave is 45°, or one-eighth period 
behind that of the wave coming from the center of the zone system (the Huygens' 
wavelet reaching P from M in Fig. 18/0 • A similar phase discrepancy, this time 
of one-quarter period, occurs in the treatment of circular zones in Sec. 18.6. The 
difference results from the fact that in the representation of a cylindrical wave, to be 
used in the Kirchhoff integral (Sec. 18.17), there occurs an additional phase constant of 
7r/4 as compared to that for a spherical wave. The result of the integration over the 



FRESNEL DIFFRACTION 



369 



18.11. The Straight Edge. The investigation of the diffraction by a 
single screen with a straight edge is perhaps the simplest application of 
Cornu's spiral. Figure 180(a) represents a section of such a screen, 
having its edge parallel to the slit S. In this figure the half-period strips 
corresponding to the point P being situated on the edge of the geometrical 
shadow are marked off on the wave front. To find the intensity at P, 
we note that since the upper half of the wave is effective, the amplitude 
is a straight line joining and Z (Fig. 18P) of length \/y/2. The square 
of this is 1/2, so that the intensity at the edge of the shadow is just one- 
fourth of that found above for the unobstructed wave. 



fif- 




(a) (ft) 

Fig. 180. Illustrating two different positions of the half-period strips relative to a 
straight edge N. 

Consider next the intensity at the point P' [Fig. 180(a)] at a distance 
I above P. To be specific, let P' he in the direction SM h where M\ is 
the upper edge of the first half-period strip. For this point, the center 
Mo of the half-period strips lies on the straight line joining S with P', and 
the figure must be reconstructed as in Fig. 180(6). The straight edge 
now lies at the point M [, so that not only all the half-period strips above 
Mo are exposed but also the first one below Mo. The resultant ampli- 
tude A is therefore represented on the spiral of Fig. 18P by a straight line 
joining B' and Z. This amplitude is more than twice that at P, and the 
intensity A 2 more than four times as great. 

Starting with the point of observation P at the edge of the geometrical 
shadow (Fig. 180), where the amplitude is given by OZ, if we move the 
point steadily upward, the tail of the amplitude vector moves to the left 
along the spiral, while its head remains fixed at Z. The amplitude will 
evidently go through a maximum at b', a minimum at c', another maxi- 
mum at d', etc., approaching finally the value Z'Z for the unobstructed 
wave. If we go downward from P, into the geometrical shadow, the tail 



entire surface is, as explained in Sec. 18.17, to bring the phase of the resultant in both 
cases into agreement with that of the direct wave. For a discussion of the phase 
discrepancy in Cornu's spiral, see It. W. Ditchbum, "Light," 1st ed., p. 214, Inter- 
science Publishers, Inc., New York, l'.)5;i. 



370 



PHYSICAL OPTICS 




Fig. 18P. Cornu's spiral, showing resultants for straight-edge diffraction pattern 




(6) 

1 


\~\ 


c' 


K^y 


p 





+2 
-v — 



-2 



-4 



-6 



Fig. 18Q. Amplitude and intensity contours for Fresnel diffraction at a straight edge. 

of the vector moves to the right from 0, and the amplitude will decrease 
steadily, approaching zero. 

To obtain quantitative values of the intensities from Cornu's spiral, 
it is only necessary to measure the length A for various values of v. The 
square of A gives the intensity. Plots of the amplitude and the intensity 



FRESNEL DIFFRACTION 



371 



against v are shown in Figs. 18Q(a) and (6), respectively. It will be seen 
that at the point 0, which corresponds to the edge of the geometrical 
shadow, the intensity has fallen to one-fourth that for large negative 
values of v, where it approaches the value for the unobstructed wave. 
The other letters correspond with points similarly labeled on the spiral, 
B', C, D' . . . , representing the exposure of one, two, three, etc., half- 
period strips below M . The maxima and minima of these diffraction 
fringes occur a little before these points are reached. For instance, the 








«'*«w( 



Fig. 18R. Straight-edge diffraction patterns photographed with (a) visible light of 
wavelength 4300 A and (6) X rays of wavelength 8.33 A. (c) Microphotometer trace 
of (a). 

first maximum at b' is given when the amplitude vector A has the posi- 
tion shown in Fig. 18P. Photographs of the diffraction pattern from a 
straight edge are shown in Fig. 187? (a) and (6). Pattern (a) was taken 
with visible light from a mercury arc, and (6) with X rays, X = 8.33 A. 
Figure 18 R(c) is a density trace of the photograph (a), directly above, 
and was made with a microphotometer. 

Perhaps the most common observation of the straight-edge pattern, 
and certainly a very striking one, occurs in viewing a distant street lamp 
through rain-spattered spectacles. The edge of each drop as it stands on 
the glass acts like a prism, and refracts into the pupil of the eye rays 
which otherwise would not enter it. Beyond the edge the field is there- 
fore dark, but the crude outline of the drop is seen as an irregular bright 
patch bordered by intense diffraction fringes such as those shown in Fig. 
18R. The fringes are very clear, and a surprising number may be seen, 
presumably because of the achromatizing effect of the refraction. 

18.12. Rectilinear Propagation of Light. When we investigate the 
scale of the above pattern for a particular case, the reason for the appar- 



372 



PHYSICAL OPTICS 



ently rectilinear propagation of light becomes clear. Let us suppose that 
in a particular case a = b = 100 cm, and X = 5000 A. From Eq. 18k, 
we then have 



s = v 



ab\ 



2(a + 6) 



= 0.0354v cm 



This is the distance along the wave front [Fig. 180(a)]. To change it to 
distances I on the screen, we note from the figure that 



a \ 2a 

For the particular case chosen, therefore, 

I = 2s = 0.0708v cm 

Now in the graph of Fig. 18Q(6) the intensity at the point v = +2 is 
only 0.025 or one-eightieth of the intensity if the straight edge were 
absent. This point has I = 0.142 cm, and therefore lies only 1.42 mm 
inside the edge of the geometrical shadow. The part of the screen below 
this will lie in practically complete darkness, and this must be due to 
the destructive interference of the secondary wavelets arriving here from 
the upper part of the wave. 

18.13. Single Slit. We next consider the Fresnel diffraction of a single 
slit with sides parallel to a narrow source slit S [Fig. 185(a)]. By 



Mo 



M 



P' 



P S 




N 



(a) (b) 

Fig. 18S. Showing division of the wave front for Frensel diffraction by a single slit. 

the use of Cornu's spiral we wish to determine the distribution of the 
light on the screen PP'. With the slit located as shown, each side 
acts like a straight edge to screen off the outer ends of the wave front. 
We have already seen in Sec. 18.11 how to investigate the pattern 
from a single straight edge, and the method used there is readily extended 
to the present case. With the slit in the central position of Fig. 185(a), 
the only light arriving at P is that due to the wave front in the inter- 



FRESNEL DIFFRACTION 



373 



val As = MN. In terms of Cornu's spiral we must now determine 
what length Ay corresponds to the slit width As. This is done by Eq. 
18fc, using Ay for v and As for s. Let a = 100 cm, b = 400 cm, X = 
4000 A = 0.00004 cm, and the slit width As = 0.02 cm. Substituting in 
Eq. 18fc, we obtain Ay = 0.5. The resultant amplitude at P is then given 




Fig. 187". Cornu's spiral, showing the chords of arcs of equal lengths Av. 

by a chord of the spiral, the arc of which has a length Ay = 0.5. Since 
the point of observation P is centrally located, this arc will start at 
v = —0.25 and run to v = +0.25. This amplitude A ~0.5 when 
squared gives the intensity at P. 

If we now wish the intensity at P' [Fig. 18<S(6)], the picture must be 
revised by redividing the wave front as shown. With the point of 
observation at P' , the same length of wave front, As = 0.02 cm, is 
exposed, and therefore the same length of the spiral, Ay = 0.5, is effec- 
tive. This section on the lower half of the wave front will, however, 
correspond to a new position of the arc on the lower half of the spiral. 
Suppose that it is represented by the arc jk in Fig. 187 7 . The resultant 



374 PHYSICAL OPTICS 

amplitude is proportional to the chord A, and the square of this gives 
the relative intensity. Thus to get the variation of intensity along the 
screen of Fig. 18S, we slide a piece of the spiral of constant length Av = 0.5 
to various positions and measure the lengths of the corresponding chords 
to obtain the amplitudes. In working a specific problem, the student 
may make a straight scale marked off in units of v to tenths, and measure 
the chords on an accurate plot such as Fig. 18iV, using the scale of 




At>=1.5 



At; =4.6 



x-rays 



-5 +5 

v 



Av = 2.5 




-4 +4 



At; = 3.9 




-*3 +3 





-6 -3 

Fig. 18 U. Fresnel diffraction of visible light by narrow slits. 
Kellstrom.) 



+3 +6 
(X-ray pattern after 



v on the spiral to obtain the constant length Ay of the arc. The results 
should then be tabulated in three columns, giving v, A, and A 2 . The 
value of v to be entered is that for the central point of the arc whose 
chord A is being measured. For example, if the interval from v = 0.9 
to v = 1.4 is measured (Fig. 182'), the average value v = 1.15 is tabu- 
lated against A = 0.43. 

Photographs of a number of Fresnel diffraction patterns for single 
slits of different widths are shown in Fig. 18C7 with the corresponding 
intensity curves beside them. These curves have been plotted by the 
use of Cornu's spiral. It is of interest to note in these diagrams the 
indicated positions of the edges of the geometrical shadow of the slit 
(indicated on the v axis). Very little light falls outside these points. 
For a very narrow slit like the first of these where Ay = 1.5, the pattern 



FRESNEL DIFFRACTION 375 

greatly resembles the Fraunhofer diffraction pattern for a single slit. 
The essential difference between the two (cf. Fig. 15D) is that here the 
minima do not come quite to zero except at infinitely large v. The small 
single-slit pattern at the top was taken with X rays of wavelength 8.33 A, 
while the rest were taken with visible light of wavelength 4358 A. As the 
slit becomes wider, the fringes go through very rapid changes, approach- 
ing for a wide slit the general appearance of two opposed straight-edge 
diffraction patterns. The small closely spaced fringes superimposed on 
the main fringes at the outer edges of the last figure are clearly seen in the 
original photograph and may be detected in the reproduction. 

18.14. Use of Fresnel's Integrals in Solving Diffraction Problems. 
The tabulated values of Fresnel's integrals in Table 18-1 may be used 
for higher accuracy than that obtainable with the plotted spiral. For 
an interval Av = 0.5, for example, the two values of x at the ends of this 
interval are read from the table and subtracted algebraically to obtain 
Ax, the horizontal component of the amplitude. The corresponding 
two values of y are also subtracted to obtain Ay, its vertical component. 
The relative intensity will then be obtained by adding the squares of 
these quantities, since 

/~4 2 = (Ax) 2 4- (Ay) 2 (18p) 

The method is accurate, but may be tedious. This is especially so if 
good interpolations are to be made in certain parts of Table 18-1. Some 
problems, such as that of the straight edge, are simplified by the fact 
that the number of zones on one end of the interval is not limited. The 
values of both x and y will be | at this end. Another example of this 
type will now be considered. 

18.15. Diffraction by an Opaque Strip. The shadow cast by a narrow 
object with parallel sides, such as a wire, may also be studied by the use of 
Cornu's spiral. In the case of a single slit, treated in Sec. 18.13, it was 
shown how the resultant diffraction pattern is obtained by sliding a 
fixed length of the spiral, Av = const., along the spiral and measuring 
the chord between the two end points. The rest of the spiral out to 
infinity, i.e., out to Z or Z' on each side of the element in question, was 
absent owing to the screening by the two sides of the slit. If now the 
opening of the slit in Fig. 18S(a) is replaced by an object of the same 
size, and the slit jaws taken away, we have two segments of the spiral 
to consider. Suppose the obstacle is of such a size that it covers an 
interval Ay = 0.5 on the spiral (Fig. 18T). For the position jk the 
light arriving at the screen will be due to the parts of the spiral from Z' 
to j and from k to Z. The resultant amplitude due to these two sections 
is obtained by adding their respective amplitudes as vectors. The lower 
section gives an amplitude represented by a straight line from Z' to j, 



376 



PHYSICAL OPTICS 



with the arrowhead at j. The amplitude for the upper section is repre- 
sented by a straight line from k to Z with the arrowhead at Z. The vector 
sum of these two gives the resultant amplitude A and A °- gives the inten- 
sity for a point v halfway between j and k. Photographs of three diffrac- 
tion patterns produced by small wires are shown in Fig. 18 V, accompanied 
by the corresponding theoretical curves. 




Vm|||IaJ 



Atf = 0.5 




-5 



+5 —V-+. 






-5 +5 — v- 

Fig. 18V. Fresnel diffraction by narrow opaque strips. 



18.16. Diffracting Screens of Other Shapes. Babinet's Principle. 
From the foregoing examples, the method should be clear for investigat- 
ing any problem, however complex, in which all edges of the screen are 
parallel to the source slit. The student will find it instructive, for exam- 
ple, to work out a double-slit pattern in Fresnel diffraction. Care must 
always be taken to get the proper direction for any individual amplitude 
vector obtained as a chord of the Cornu spiral. Since the spiral is made 
up of infinitesimal amplitude vectors starting at Z' and ending at Z 
(Sec. 18.8), the arrowhead of any vector must be at the end nearer 
on the spiral to Z. 

Screens having straight edges which are not parallel, as, for example, a 
triangle or a polygon, will not produce clear diffraction patterns unless 
illuminated by a point source. Cornu's spiral is therefore not applicable 



FRESNEL DIFFRACTION 377 

to them, and recourse must be had to the more general theory mentioned 
below. A striking feature of such patterns is the appearance of light fans 
spreading out in both directions perpendicular to any straight edge of the 
screen.* Thus a point source when viewed through a small opening 
in the shape of an equilateral triangle looks like a six-pointed star. 

There is a generalization known as Babinet's principle which relates 
the diffraction patterns produced by two complementary screens. The 
term complementary here signifies that the opaque spaces in one screen 
are replaced by transparent spaces in the other, and vice versa. An 
opaque strip, for example, is complementary to a slit of the same width. 
In its most general form, the principle states that the vector amplitude 
produced at a given point by one screen, when added to that produced 
by the other screen, gives the amplitude due to the unscreened wave. In 
effect it says that the whole is the sum of its parts. Thus we may write 
the vector equation 

A, + A 2 = A (18q) 

where the subscripts 1 and 2 refer to the complementary screens, and 
to the absence of any screen. The truth of the principle may be verified 
with Cornu's spiral by dividing it into the appropriate parts, although it is 
applicable to all other types of diffraction as well. 

Babinet's principle is not very useful in dealing with Fresnel diffrac- 
tion, except as it may furnish a short-cut in obtaining the pattern for a 
particular screen from that of its complement. In Fraunhofer diffraction, 
however, it has an interesting consequence. Here the unscreened wave 
yields intensity zero over the whole field except at the image of the source 
itself. Thus A = 0, and we have A 2 = — Ai. When these amplitudes 
are squared to obtain the intensities, we find that the diffraction patterns 
due to complementary screens are identical. That this statement is far 
from true in a typical case of Fresnel diffraction may be seen by compar- 
ing Figs. 18U and 18 V. It would apply to the case of a fine wire stretched 
over the objective of an astronomical telescope, where there would be 
produced in the image plane a faint single-slit pattern in the corresponding 
orientation. Finally, it should be mentioned that Babinet's principle 
is not perfectly rigorous, but involves approximations,! as does the rest 
of the Huygens-Fresnel treatment thus far discussed. 

18.17. More General Treatments of Diffraction. The original appli- 
cation of Huygens' principle to diffraction, although it gave results 

* Excellent photographs of diffraction patterns due to openings of various shapes 
will be found in G. Z. Dimitroff and J. G. Baker, "Telescopes and Accessories," 
appendix VIII, The Blakiston Division, McGraw-Hill Book Company, Inc, New York, 
1945. 

fSee E. T. Copson, Proc. Roy. Soc. (London), 186, 116, 1946, where its limits of 
applicability are discussed. The principle is not even approximately true in the case 
of a perfectly reflecting screen. 



378 PHYSICAL OPTICS 

agreeing with experiment in problems like those discussed earlier in this 
chapter, contained certain assumptions which were definitely incorrect. 
Fresnel took the obliquity factor as cos 0, since it seemed reasonable 
that the surface element of the wave might radiate according to Lam- 
bert's law. He neglected the fact that the phase of the resultant wave 
came out wrong, since he was interested only in predicting intensities. A 
more important question, however, is whether an error is not made in 
assuming uniform amplitude and phase over the wave front in the clear 
part of the diffracting screen, and zero amplitude behind the opaque 
parts. The more refined mathematical theories* developed since Fres- 
nel's time have given the correct answers to these questions, and have 
also shown where the limitations of his method lie. 

The first important advance was made by Kirchhoff in 1876, who 
showed that the light wave at any point in space could be expressed as an 
integral over a closed surface surrounding that point. The Huygens 
secondary wavelets appear in this theory as the differential contributions 
from the surface elements, and when part of the surface coincides with 
the wave front, their amplitude is found to vary as 1 + cos 6, as we 
assumed in Sec. 18.1. Furthermore, when the integration is extended 
over the entire surface, one obtains the wave exactly as it would have 
reached the point directly from the source, i.e., with its correct amplitude 
and phase. Hence two deficiencies of Fresnel's theory, relating to the 
obliquity factor and to the phase, are supplied by Kirchhoff's extension. 

In principle, the solution of any diffraction problem may be obtained 
by making part of Kirchhoff's closed surface coincide with the diffracting 
screen, and evaluating his integral with suitable boundary conditions. 
To do this, however, one needs to know the values of the complex ampli- 
tude, and of its derivative with respect to the normal, over the whole 
surface. Actually, these are never accurately known, and to solve the 
problem certain simplifying assumptions must be made, which in the end 
yield little more than the original Fresnel treatment. The results are 
in fact identical when the apertures are many wavelengths wide, and the 
observations are made at any appreciable distance from them. Recently, 
it has become possible by the use of microwaves of a few centimeters 
wavelength to measure diffraction patterns right up to the plane of the 
aperture, and with apertures of width from several wavelengths to only a 
fraction of the wavelength, f The results show surprisingly good agree- 
ment with the approximate Kirchhoff theory, but also indicate the need 

* A complete and authoritative account will be found in A. Sommcrfeld, "Optics," 
chaps. 5 and 6, Academic Press, Inc., New York, 1954. For a good summary from a 
more elementary standpoint, see J. Valasek, "Theoretical and Experimental Optics," 
pp. 172-186, John Wiley & Sons, Inc., New York, 1949. 

t C. L. Andrews, J. Appl. Phys., 21, 761, 1950; Am. J. Phys., 19, 280, 1951. 



FRESNEL DIFFRACTION 379 

for further theoretical and experimental studies of diffraction by these 
methods. Since the Fresnel approach is adequate in considering diffrac- 
tion for optical wavelengths, however, we shall not take up these inter- 
esting developments. 

After the advent of the electromagnetic theory of light, attempts were 
made to obtain rigorous treatments of certain simple types of diffraction 
by applying specific boundary conditions to Maxwell's equations (Chap. 
20). These conditions involve a knowledge of the electrical properties 
of the material of the diffracting screen itself. Sommerfeld was successful 
in solving the one problem of the straight edge, for a screen of infinitesimal 
thickness and perfect reflectance, by this method. An interesting point 
came out of this work, which explained an observation that had long 
puzzled those who studied diffraction experimentally. When the eye is 
placed in the region of the diffracted light, the diffracting edge or edges 
appear luminous, even though precautions are taken to avoid reflected 
or scattered light. Sommerfeld's theory derives the resultant wave 
arriving at a point on the screen in all detail, including its phase distribu- 
tion. In the geometrical shadow of the straight edge, it is found to 
consist of a cylindrical wave apparently originating at the edge. Outside 
of the shadow there are both the direct wave and this deflected one, and 
the diffraction fringes observed there can be interpreted as due to the 
interference of these two. This is, in fact, the original explanation of 
diffraction fringes given by Thomas Young, an explanation which was 
until recently regarded as erroneous. It constitutes an alternative 
interpretation, and one which is mathematically equivalent to that of 
Kirchhoff, of any Fresnel diffraction phenomenon. The single-slit pat- 
tern may be regarded as caused by the interference of the direct wave 
and two cylindrical waves, one from each edge.* 

PROBLEMS 

1. Write out the series of terms analogous to those in Eq. 18/ when the number of 
zones is even, and show that in that case the resultant amplitude is (Ai/2) — (.Am/2). 

2. Carry the calculation of the areas of the Fresnel zones to a higher order of 
approximation than that of Eq. 18c. For a plane wave front, by what percentage 
does the area of the thirtieth zone exceed that of the first? Take X = 5000 A and 
b = 20 cm. Ans. 0.0037%. 

3. Make polar plots of the amplitude and intensity of a Huygens secondary wavelet 
against 0, using the correct obliquity factor. What is the name for the mathematical 
curve giving the amplitude? 

* Nonmathematical discussions of the luminosity of the diffracting edge and of 
Young's theory will be found in C. F. Meyer. "The Diffraction of Light, X-rays and 
Material Particles," 1st ed., chap' 7, sees. 10-11, University of Chicago Press, Chicago, 
1934, and in R. W. Wood, "Physical Optics," 3d ed., pp. 218-221, The Macmillan 
Company, New York, 1933. 



380 PHYSICAL OPTICS 

4. One of the zone plates of Fig. 1 8/ is photographed on such a scale that the first 
zone, as measured on a comparator, has a radius of 0.390 mm. It is then mounted 
on an optical bench 42 cm from a pinhole illuminated by the green mercury line, 
X5461. Find the distance from the zone plate of the primary image, and also of the 
first two subsidiary ones. Ans. 82.7. 11.9. 6.4 cm. 

6. A parallel beam of microwaves having X = 3 cm passes through a circular opening 
of adjustable radius. If a detector is placed on the axis of the hole 4 m behind it 
and the opening gradually increased in radius, at what value would the response 
reach its first maximum? Its second minimum? At the latter radius, give an equa- 
tion for the positions of the maxima and minima along the axis. 

6. Assume that the bright spot in the shadow of a disk is visible when the deviations 
from a perfectly circular contour do not exceed one-third of the width of a zone. If a 
1-cent piece 18.5 mm in diameter is placed in red light (6000 A.) from a distant point 
source, and its shadow viewed by an eyepiece 1 m behind it, what is the maximum 
allowable variation of the radius? Ans. 0.01 1 mm. 

7. When a star is eclipsed by the moon, find how long a time it would take for the 
intensity to fall to one-hundredth of its initial value. 

8. Using Cornu's spiral, plot the diffraction pattern of a single slit having a width 
As = 1.2 mm. Assume a = 100 cm, 6 = 150 cm, X = 5000 A. 

Ans. Plot, with three strong maxima of almost equal intensity, and weaker side 
maxima. 

9. A slit is placed at one end of an optical bench, and is illuminated with sodium 
light. A holder for diffracting objects is located 60 cm from the slit, and observations 
are made with a photoelectric cell behind a narrow slit 120 cm from the object holder. 
What would be the exact intensity, relative to the unobstructed intensity, (a) at the 
edge of the geometrical shadow of a rod 1.8 mm thick, (ft) at the center of the shadow 
of this rod? 

10. In the arrangement of Prob. 9, find the intensity (a) 2 mm inside the edge 
of the geometrical shadow of a straight edge, (6) 1 mm outside this edge. 

Ans. (a) 0.013 h; (b) 1.23 I . 

11. In the arrangement of Prob. 9, find the intensity (a) at one edge of the geo- 
metrical shadow of a single slit 1.5 mm wide, (b) at the center of the pattern of a 
single slit 2.5 mm wide. 

12. From the table of Fresnel's integrals, calculate the exact intensity at the points 
v = +1.2 and —2.0 in the diffraction pattern of a straight edge. To what angles of 
diffraction do these correspond when the source is very distant and the wavelength is 
(a) 4000 A, and (ft) 5 cm? The observing screen is 5 m behind the straight edge. 

Ans. 0.0308/o, 0.844/ . (a) 0.014°, -0.023°. (ft) 4.85°, -8.05°. 

13. Use Cornu's spiral to investigate the Fresnel diffraction pattern of a double 
slit. If a = b = 100 cm, X = 5000 A, and the slits are each 0.4 mm wide with their 
centers 2 mm apart, find the distance from the center of the pattern of (a) the first 
minimum, (ft) the second maximum. 

14. Derive, by using Babinet's principle, a simple relation between the intensity 
at a point in the single-slit diffraction pattern, and the intensity at the same point 
in the pattern due to the complementary opaque strip. 

Ans. (///o)stri P = (///o)siit — Ax — Ay -\- 1. Here Ax and Ay are the components 
of the amplitude vector for the slit. 

15. For the diffraction by an opaque strip, investigate by Cornu's spiral (a) whether 
a maximum must necessarily occur at the center of the pattern, as it does in the three 
cases of Fig. 18F; (ft) the origin of the "beats" observed outside the geometrical 
shadow in the case At> = 0.5 of Fig. 187. 



FRESNEL DIFFRACTION 381 

16. According to Young's interpretation of diffraction, the fringes inside the geo- 
metrical shadow of an opaque strip are to be regarded as interference fringes caused 
by the two luminous edges. On this hypothesis, how many bright fringes should 
occur in the shadow of an opaque strip of width Av = 2.5 (cf. Fig. 187)? In the 
shadow of the rod of Prob. 9? Ans. 3 fringes. 13 fringes. 

17. A plane wave of light of wavelength 5000 A is incident on a screen containing 
a circular hole 1 mm in diameter. Find the intensity on the axis at a distance of 
30 cm behind the screen, expressing it as a fraction of the intensity at the first maxi- 
mum. The latter is the intensity at the point where the hole just includes the first 
Fresnel zone. 



CHAPTER 19 

THE VELOCITY OF LIGHT 



In the preceding chapters we have found that the interference and 
diffraction of light can be successfully explained by assuming that light 
consists of waves. We now turn to another fundamental property of 
light waves, their velocity of propagation. It is to be expected that 
waves having a definite frequency will travel with finite and constant 
velocity in a given medium. Light waves, or in general, electromagnetic 
waves, are unique in their ability to move through empty space and 
here the velocity is the same for all frequencies. Hence the velocity of 
light in vacuum, c, is an important constant of nature. In the electro- 
magnetic theory, to be discussed in the next chapter, it appears as the 
ratio of certain units. Furthermore, the discovery that its observed value 
is independent of motion of either source or observer formed the original 
basis of the theory of relativity. Our first object will be to describe the 
various ways in which this characteristic velocity has been accurately 
measured. 

19.1. Romer's Method. Because of the very great velocity of light, 
it is natural that the first successful measurement was an astronomical 
one, because here very large distances are involved. In 1G76 Romer* 
studied the times of the eclipses of the satellites of the planet Jupiter. 
Figure 19 A(a) shows the orbits of the earth and of Jupiter around the 
sun S and that of one of the satellites M around Jupiter. The inner 
satellite has an average period of revolution T = 42 hr 28 min 16 sec, 
as determined from the average time between two passages into the 
shadow of the planet. Actually Romer measured the times of emergence 
from the shadow, while the times of transit of the small black spot 
representing the shadow of the satellite on Jupiter's surface across the 
median line of the disk can be still more accurately measured. 

A long series of observations on the eclipses of the first satellite per- 
mitted an accurate evaluation of the average period T . Romer found 
that if an eclipse was observed when the earth was at such a position 
as Ei [Fig. 19 A (a)] with respect to Jupiter J x , and the time of a later 

* Olaf Romer (1644-1710). Danish astronomer. His work on Jupiter's satellites 
was done in Paris, and later he was made astronomer royal of Denmark. 

382 



THE VELOCITY OF LIGHT 



383 



eclipse was predicted by using the average period, it did not in general 
occur at exactly the predicted time. Specifically, if the predicted eclipse 
was to occur about 3 months later, when the earth and Jupiter were 
at E 2 and J 2 , he found a delay of somewhat more than 10 min. To 
explain this, he assumed that light travels with a finite velocity from 
Jupiter to the earth, and that since the earth at E 2 is farther away from 
Jupiter, the observed delay represents the time required for light to 
travel the additional distance. His measurements gave 11 min as the 
time for light to go a distance equal to the radius of the earth's orbit. 




Fig. 19 A. Principle of Romer's astronomical determination of the velocity of light 
from observations on Jupiter's moons. 

We now know that 8 min 18 sec is a more nearly correct figure, and com- 
bining this with the average distance to the sun 93 X 10 6 miles, we find 
a velocity of about 187,000 mi/sec. 

It is instructive to inquire how the apparent period of the satellite, 
i.e., the time between two successive eclipses, is expected to vary through- 
out a year. If this time could be observed with sufficient accuracy, one 
would obtain the curve of Fig. 19-4.(6). We may regard the successive 
eclipses as light signals sent out at regular time-intervals of 42 hr 28 min 
16 sec from Jupiter. Now at all points in its orbit except E x and E» 
the earth is changing its distance from Jupiter more or less rapidly. If 
the distance is increasing, as at E 2 , any one signal travels a greater dis- 
tance than the preceding one and the observed time between them will 
be increased. Similarly at E A it will be decreased. The maximum 
variation from the average period, about 15 sec, is the time for light to 
cover the distance moved by the earth between two eclipses, which 
amounts to 2.8 X 10" miles. At any given position, the total time delay 
of the eclipse, as observed by Romer, will be obtained by adding the 



384 



PHYSICAL OPTICS 



Solar 
system 



Observers 
telescope 



(61 



amounts T — T [Fig. 19 A (b)], by which each apparent period is longer 
than the average. For instance, the delay of an eclipse at E 2 , predicted 
from one at Ei using the average period, will be the sum of T — T for 
all eclipses between Ei and E 2 . 

19.2. Bradley's* Method. The Aberration of Light. Romer's inter- 
pretation of the variations in the times of eclipses of Jupiter's satellites 
was not accepted until an entirely independent determination of the 
velocity of light was made by the English astronomer Bradley in 1727. 

Bradley discovered an apparent 
motion of the stars which he ex- 
plained as due to the motion of 
the earth in its orbit. This effect, 
known as aberration, is quite dis- 
tinct from the well-known dis- 
placements of the nearer stars 
known as parallax. Because of 
parallax, these stars appear to 
shift slightly relative to the back- 
ground of distant stars when they 
are viewed from different points 
in the earth's orbit, and from 
these shifts the distances of the 
stars are computed. Since the 
apparent displacement of the star 
is opposite to that of the position 
of the earth, the effect of paral- 
lax is to cause the star which is observed in a direction perpendicular 
to the plane of the earth's orbit to move in a small circle with a phase 
differing by tt from the earth's motion. The angular diameters of 
these circles are very small, being not much over I second of arc for 
the nearest stars. Aberration, which depends on the earth's velocity, 
also causes the stars observed in this direction apparently to move 
in circles. Here, however, the circles have an angular diameter of about 
41 seconds, and they are the same for all stars, whether near or distant. 
Furthermore, the displacements are always in the direction of the earth's 
velocity, so that the circular motions are ir/2 different in phase from the 
earth's motion [Fig. 195(a)]. 

Bradley's explanation of this effect was that the apparent direction of 
the light reaching the earth from a star is altered by the motion of the 
earth in its orbit. The observer and his telescope are being carried 

* James Bradley (1693-1762). At the time professor of astronomy at Oxford. He 
got his ideas about aberration by a chance observation of the changes in the apparent 
direction of the wind while sailing on the Thames. 




Light 

Fig. 192?. Origin of astronomical aber- 
ration, when the star is observed per- 
pendicular to the plane of the earth's 
orbit. 



THE VELOCITY OF LIGHT 385 

along with the earth at a velocity of about 18.5 mi/sec, and if this motion 
is perpendicular to the direction of the star, the telescope must be 
tilted slightly toward the direction of motion from the position it would 
have if the earth were at rest. The reason for this is much the same 
as that involved when a person walking in the rain must tilt his umbrella 
forward to keep the rain off his feet. In Fig. 19B(b), let the vector v 
represent the velocity of the telescope relative to a system of coordinates 
fixed in the solar system, and c that of the light relative to the solar 
system. We have represented these motions as perpendicular to each 
other, as would be the case if the star lay in the direction shown in Fig. 
19J5(a). Then the velocity of the light relative to the earth has the 
direction of c', which is the vector difference between c and v. This is 
the direction in which the telescope must be pointed to observe the star 
image on the axis of the instrument. We thus see that when the earth 
is at Ei the star S has the apparent position Si, when it is at Ei, the 
apparent position is Si, etc. If S were not in a direction perpendicular 
to the plane of the earth's orbit, the apparent motion would be an ellipse 
rather than a circle, but the major axis of the ellipse would be equal to 
the diameter of the circle in the above case. 

It will be seen from the figure that the angle a, which is the angular 
radius of the apparent circular motion, or the major axis of the elliptical 
one, is given by 

tan a «■» - (19a) 

c 

Recent measurements of this so-called angle of aberration give a mean 
value a = 20.479" ± 0.008 as the angular radius of the apparent circular 
orbit. Combining this with the known velocity v of the earth in its 
orbit, we obtain c = 186,233 mi/sec, or 299,714 km/sec. This value 
agrees to within its experimental error with the more accurate results 
obtained by the latest measurements of the velocity of light by direct 
methods, the principles of which we shall now describe. 

19.3. Fizeau's Terrestrial Method. Fizeau,* in 1849, first succeeded 
in measuring c by a method not involving astronomical observations, 
i.e., one in which the light path was on the earth's surface. The prin- 
ciple of his determination was the obvious one of sending out a brief 
flash of light and measuring the time for this to travel to a distant mirror 
and back to the observer. This was accomplished with the apparatus 
shown in Fig. 19C. The cogwheel WF is rotated at high speed so that 
it cuts the light beam passing through the rim at F into a series of short 

* H. L. Fizeau (1819-1896). Born of a wealthy French family, he was financially 
independent to pursue his hobby — the velocity of light. His experiments were carried 
out in Paris, the light traveling between Montmartre and Suresnes. 



386 



PHYSICAL OPTICS 



flashes. A flash is sent out each time the wheel is in such a position 
that the light can pass between two cogs. It is then rendered parallel 
by the lens L 2 and focused by L 3 on a plane mirror M. In Fizeau's 
experiments the distance MF was 5.36 miles. After reflection from M, 
the flash of light retraces its path, and is again focused by L 2 on the rim 
of the wheel. If during the time that the light has traveled from F to 
M and back the wheel has turned to such a position that a cog is inter- 
posed at F, this flash will be cut out, and the same mil be true of any 
other flash. 




Fig. 19C. Fizeau's experimental arrangement used in the first terrestrial determination 
of the velocity of light. 

With the wheel at rest in such a position that the light traverses the 
opening between two cogs (Fig. 19C, center), the observer at E will 
see the image of the light source at F by means of the eyepiece L 4 , focused 
on F through the half-silvered mirror G. If the wheel is now rotated 
with increasing speed, a state will be reached in which the light passing 
is stopped by a, that passing 1 is stopped by cog b, etc., and the image 
will be completely eclipsed. A further increase in speed will cause the 
light to reappear when these flashes pass through openings 1, 2, ... , 
and a second eclipse will occur where they are stopped by b, c, ... . 
Fizeau's wheel had 720 cogs, and since the light path was 2 X 5.36 or 
10.72 miles, the wheel had to turn through y^ of a revolution in 10.72/c 
sec to produce the first eclipse. Hence the first eclipse should occur at a 
speed of c/(10.72 X 1440) rev/sec, and the others at 3, 5, 7, . . . times 
this speed. Fizeau observed the first eclipse at 12.6 rev/sec, giving 
c = 194,600 mi/sec or 313,300 km/sec. 

That this is appreciably higher than the values obtained by the astro- 
nomical methods is not surprising, in view of the difficulties of the 
experiment. With Fizeau's arrangement, the determination of the exact 



THE VELOCITY OF LIGHT 



387 



condition of total eclipse caused the principal uncertainty. The experi- 
mental conditions were later improved by Cornu, and by Young and 
Forbes. The latter overcame the above difficulty by placing another 
lens and mirror, identical with L 3 and M, at a somewhat greater distance. 
The two images thus formed were observed simultaneously, and instead 
of measuring the conditions of eclipse or of maximum in either image 
they measured the speed of the cogwheel at the time the two images 
appeared to be of equal intensity. The eye is very sensitive to the 
detection of slight differences in intensity of adjacent images, so this 




Fig. 19D. Rotating-mirror apparatus used by Foucault in measuring the velocity of 
light. 

measurement could be made more accurately. Their result* was 
301,400 km/sec. 

19.4. Rotating-mirror Method. This is a second terrestrial method, 
originally suggested by Aragof and first applied successfully by Fizeau 
and Foucaultt independently in 1850. The principle of these early 
determinations is illustrated in Fig." 19D. Light from the source S 
traverses the plane glass plate G and, after reflection from the plane 
mirror R, is focused by the lens L on a stationary concave mirror M. 
If R is also stationary, the light retraces its path and an image of S is 
formed at E by partial reflection in G. 

If now R is rotated at high speed about an axis perpendicular to the 

* For details of the various determinations by Fizeau's method, the reader is referred 
to T. Preston, "The Theory of Light," 5th ed., p. 534, Macmillan & Co., Ltd., London, 

1928. 

t D. F. J. Arago (1786-1853). Noted Parisian astronomer and physicist. He is 
principally known for his work on the interference of polarized light (Chap. 27) and 
on electromagnetism in conjunction with Ampere. 

% J. L. Foucault (1819-1868). Between 1845 and 1849 Foucault collaborated with 
Fizeau, but owing to difference of opinion they afterward worked separately. Fou- 
cault is also known for his demonstration of the rotation of the earth by a pendulum 
and for the Foucault knife-edge test. His researches on the velocity of light in water 
(Sec. 19.10) constituted his thesis for the doctorate, presented in 1851. 



388 PHYSICAL OPTICS 

plane of the figure, it will have turned through a small angle a by the 
time the light has returned from M. The reflected beam will then be 
turned through 2a, and a displaced image E' will be produced by L. 
The displacement EE' obviously depends on the angular velocity of R 
and on the distances RM and RGE, and if these quantities are known the 
velocity of light may be found. 

In the final measurements of Foucault, RM was 20 m and essentially 
equal to the radius of curvature LM of the mirror M . The displacement 
EE' was only 0.7 mm but could be measured by the micrometer eyepiece 
to within 0.005 mm. Foucault's result for the velocity of light was 
roughly 298,000 km/sec. The accuracy of the determination by the 
rotating-mirror method was later greatly improved in the experiments 
of Cornu, Newcomb,* and Michelson. The chief improvement in the 
later work lay in the use of a greater light path. This was limited in 
Foucault's arrangement by the loss of intensity in the image when the 
distance RM was made large. The rotating beam from R is returned 
by M only during the small fraction of the time that it is sweeping 
across M. This difficulty was overcome in Michelson's work by using a 
lens L of larger focal length, and increasing the distance RL until R and 
M were nearly conjugate foci of L. With S fairly close to R, and a lens L 
of sufficiently long focus, the mirror M could now be placed several miles 
away. Another improvement adopted by Newcomb and Michelson was 
the replacement of the plane mirror R by one having four or more reflect- 
ing faces (Fig. 19E). This also resulted in a gain of intensity in the 
image, f 

19.5. Michelson's Later Experiments. We shall not describe the 
successive experiments in Avhich the determination of c by rotating mirrors 
was steadily improved. At the present time, it appears that the accuracy 
of even the best values by this method is surpassed by that of newer 
devices based on radio-frequency techniques. It will be instructive, 
however, to consider briefly a classical series of measurements made by 
Michelson at the Mt. Wilson Observatory in 1926. 

The form of the apparatus finally adopted is shown in Fig. 192?. Light 
from a Sperry arc S passes through a narrow slit and is reflected from 
one face of the octagonal rotating mirror R. Thence it is reflected from 
the small fixed mirrors b and c to the large concave mirror M i (30-ft 
focus, 2-ft aperture). This gives a parallel beam of light, which travels 
22 miles from the observing station on Mt. Wilson to a mirror M 2 , similar 
to M h on the summit of Mt. San Antonio. M 2 focuses the light on a 

* Simon Newcomb (1835-1909). Distinguished American astronomer, associated 
with the U.S. Naval Observatory and the Johns Hopkins University. 

t For further discussion of these methods, see N. E. Dorsey, Trans. Am. Phil. Soc, 
34, 1, 1944. 



THE VELOCITY OF LIGHT 389 

small plane mirror /, whence it returns to M i and, by reflection from 
c' , b', a', and p, to the observing eyepiece L. 

Various rotating mirrors, having 8, 12, and 16 sides, were used, and 
in each case the mirror was driven by an air blast at such a speed that 
during the time of transit to M 2 and back (0.00023 sec) the mirror turned 
through such an angle that the next face was presented at a'. For an 
octagonal mirror, the required speed of rotation was about 528 rev/sec. 
The speed was adjusted by a small counterblast of air until the image 
of the slit was in the same position as when U was at rest. The exact 
speed of rotation was then found by a stoboscopic comparison with a 



/ 




L 


f 




urn b ' 




R>»/ 








9 - 




Mt. San Antonio 


22 miles 








Mt. Wilson 



Fia. 19£. Michelson's arrangement used for determining the velocity of light (1926). 

standard electrically driven tuning fork, which in turn is calibrated 
with an invar pendulum furnished by the U.S. Coast and Geodetic 
Survey. This Survey also measured the distance between the mirrors 
M\ and M 2 with remarkable accuracy by triangulation from a 40-km 
base line, the length of which was determined to an estimated error of 
1 part in 11 million, or about | in.* 

The results of the measurements published in 1926 comprised eight 
values of the velocity of light, each the average of some 200 individual 
determinations with a given rotating mirror. These varied between the 
extreme values of 299,756 and 299,803 km/sec and yielded the average 
value of 299,796 + 4 km/sec. Michelson also made some later measure- 
ments with the distant mirror on the summit of a mountain 82 miles 
away, but because of bad atmospheric conditions, these were not con- 
sidered reliable enough for publication. 

19.6. Measurements in Vacuum. In the preceding discussion we have 
assumed that the measured velocity in air is equal to that in a vacuum. 
That is not exactly true, since the index of refraction n = c/v is slightly 
greater than unity. With white light the effective value of n for air 
under the conditions existing in Michelson's experiments was 1.000225. 
Hence the velocity in vacuum c = ni> was 67 km/sec greater than v, 
the measured value in air. This correction has been applied in the final 

* W. Bowie, Astrophys. J., 66, 14, 1927. 



390 PHYSICAL OPTICS 

results quoted above. A difficulty which becomes important where 
measurements as accurate as those of Michelson are concerned is the 
uncertainty of the exact conditions of temperature and pressure of the 
air in the light path. Since n depends on these conditions, the value of 
the correction to vacuum also becomes somewhat uncertain. 

To eliminate this source of error, Michelson in 1929 undertook a 
measurement of the velocity in a long evacuated pipe. The optical 
arrangement was similar to that described above, with suitable modifica- 
tions for containing the light path in the pipe. The latter was 1 mile 
long, and by successive reflections from mirrors mounted at either end 
the total distance the light traversed before returning to the rotating 
mirror was about 10 miles. A vacuum as low as $■ mm Hg could be 
maintained. This difficult experiment was not completed until after 
Michelson's death in 1931, but preliminary results were published a year 
later by his collaborators.* The mean of almost three thousand indi- 
vidual measurements was 299,774 km/sec. Because of certain unex- 
plained variations, the accuracy of this result is difficult to assess. It is 
certainly not as great as that indicated by the computed probable error, 
and has recently been estimated as ± 11 km/sec. 

19.7. Kerr-cell Method. Determinations by this method have equaled 
if not surpassed the accuracy of those by the rotating mirror. In 1925 
Gaviola devised what amounts to an improvement on Fizeau's toothed- 
wheel apparatus. It is based on the use of the so-called electro-optic 
shutter. This device is capable of chopping a beam of light several 
hundred times more rapidly than can be done by a cogwheel. Hence a 
much shorter base line can be used, and the entire apparatus can be con- 
tained in one building so that the atmospheric conditions are accurately 
known. Figure 19F(a) illustrates the electro-optic shutter, which con- 
sists of a Kerr cell K between two crossed nicol prisms N x and N*. if is a 
small glass container fitted with sealed-in metal electrodes and filled with 
pure nitrobenzene. Although the operation of this shutter depends on 
certain properties of polarized light to be discussed later (Chap. 29), all 
that need be known here in order to understand the method is that no 
light is transmitted by the system until a high voltage is applied to the 
electrodes of K. Thus by using an electrical oscillator which delivers a 
radio-frequency voltage, a light beam can be interrupted at the rate of 
many millions of times per second. 

The first measurements based on this principle used two shutters, one 
for the outgoing and one for the returning light. Except for the shorter 
distances, the method closely resembled Fizeau's. Subsequent improve- 
ments have led to the apparatus shown in Fig. 19^(6), which was used 

* The final report will be found in Michelson, Pease, and Pearson, Astrophvs. J., 
82, 26, 1935. 



THE VELOCITY OF LIGHT 



391 



by W. C. Anderson in 1941.* To avoid the difficulty of matching the 
characteristics of two Kerr cells, he used only one and divided the trans- 
mitted light pulses into two beams by means of the half-silvered mirror 
Mi. One beam traversed the shorter path to M» and back through Mi to 
the detector P. The other traveled a longer path to M 6 by reflections 
at M 3 , M 4 , and M 6 , then retraced its course to Mi which reflected it to 
P as well. This detector P was a photomultiplier tube, which responded 
to the sinusoidal modulation of the light waves. One may think of the 




M 2 M' z 




Fig. 19F. Anderson's method of measuring the velocity of light, (a) Electro-optit 
shutter, (b) The light paths. 

light wave as the carrier wave, which is amplitude-modulated at the 
frequency of the oscillator driving the Kerr cell.f The quotient of the 
wavelength I of the modulation by the period T of the oscillator thus 
gives the velocity of light. 

The accurate measurement of I is based on the following principle: 
If the longer path exceeds the shorter one by a half-integral multiple of I, 
the sum of the two modulated waves reaching P will give a constant 
intensity. The amplifier connected to the photocell was arranged to 
give zero response under this condition. The adjustment is made by 
slight motions Ay of the mirror M 2 . The extra path beyond M 4 could 
then be cut out by substituting another mirror M\ which returned the 
light directly to M 3 . If this extra path (Ma to M 6 and back) were exactly 

* /. Opt. Soc. Am., 31, 187, 1941. 

t Since the shutter transmits at each voltage peak, whether positive or negative, 
one would expect to use 1/2T here. Actually Anderson applied a d-c bias to the cell 
so that each cycle gave a single voltage maximum. 



392 PHYSICAL OPTICS 

a whole number times I, no change in the photocell response would be 
observed upon cutting it out. As the apparatus was arranged, this was 
very nearly so, the extra path being about 11/. By measuring the dis- 
placement Ar/ necessary to reestablish zero response, and applying a 
correction As involved in the substitution of M' if the difference from 112 
of the measured distance could be exactly determined. 

The reader will see the resemblance of Anderson's apparatus to a 
Michelson interferometer for radio waves, since the light pulses have a 
length essentially equal to the wave length of the radio waves given by the 
Kerr-cell oscillator. It is not exactly equal, however, because the 
velocity involved in the experiment is the group velocity of light in air, 
and not the velocity of radio waves. In his final investigation, Anderson 
made a total of 2895 observations, and the resulting velocities l/T, after 
correction to vacuum, yielded an average of 299,776 + 6 km/sec. The 
chief source of error was in the difficulty of ensuring that both beams 
used the same portion of the photoelectric surface. A change in the 
position of the light spot affects the time of transit of the electrons 
between the electrodes of the photomultiplier tube. The uncertainty 
involved here was larger than any errors in the length measurements, and 
in the frequency of the oscillator was known more accurately still — to 
better than 1 part in 1 million. 

In the latest Kerr cell determination by Bergstrand (see Table 19-1) 
the last-mentioned difficulty is avoided by using only one beam, and 
locating the maxima and minima through modulation of the detector 
in synchronism with the source. The result is indicated to be more than 
ten times as accurate as any previous one by optical methods. It dis- 
agrees with the concordant values of Anderson and of Michelson, Pease, 
and Pearson, seeming to show that Michelson's 1926 value was the more 
nearly correct. It is difficult to understand how the very thorough work 
in the period 1930-1940 could have been so far in error, but other recent 
results, to be described below, certainly put the weight of the evidence 
in favor of the higher value of c. 

19.8. Velocity of Radio Waves. The development of modern radar 
techniques, and especially the interest in their practical application as 
navigational aids, has led to renewed attempts to improve our knowledge 
of the velocity of light. This velocity is of course the same as that for 
radio waves, when both are reduced to vacuum. There are three methods 
for using microwaves for an accurate measurement of their velocity, one 
of which may easily be performed in vacuum. This is to find the length 
and resonant frequency of a hollow metal cylinder, or cavity resonator. 
It is analogous to the common laboratory method for the velocity of 
sound. Measurements of this type were made independently in Eng- 



THE VELOCITY OF LIGHT 



393 



land, by Essen and Gordon-Smith, and in America, by Bol.* As will be 
seen from Table 19-1, the results agree with each other and with Berg- 
strand's precise optical value. 

Table 19-1. Results of Accurate Measurements of the Velocity of Light 



Date 


Investigators 


Method 


Result, km /sec 


1926 


Michelson 


Rotating mirror 


299,796 + 4 


1935 


Michelson, Pease, and 
Pearson 


Rotating mirror in vacuum 


299,774 ± 11 


1940 


Huttel 


Kerr cell 


299,768 + 10 


1941 


Anderson 


Kerr cell 


299.776 ± 6 


1950 


Bol 


Cavity resonator 


299,789.3 ± 0.4 


1950 


Essen 


Cavity resonator 


299.792.5 ± 3.0 


1951 


Bergstrand 


Kerr cell 


299,793.1 +0.2 


1951 


Aslakson 


Radar (shoran) 


299,794.2 ± 1.9 


1952 


Froome 


Microwave interferometer 


299,792.6 ± 0.7 



The other methods involving radio waves are responsible for the last 
two entries in our table, and have been developed to a comparable 
accuracy. The radar method consists in the direct measurement of the 
time of transit of a signal over a known distance in the open air. The 
microwave interferometer is the Michelson instrument adapted to radio 
waves. The velocity is found by measuring the wavelength from the 
motion of a mirror. The details of all the radio methods are interesting 
and important, but must be omitted here as not falling strictly within 
the scope of optics. 

19.9. Ratio of the Electrical Units. As we shall find in our considera- 
tion of the electromagnetic theory (Chap. 20), c may be found from the 
ratio of the magnitude of certain units in the electromagnetic and electro- 
static systems. Two careful measurements of the ratio have been made, 
and have given results more or less intermediate between the higher and 
lower values discussed above. Since the accuracy thus far attained is 
considerably lower than for other methods, these experiments, although 
they have served to verify the theoretical prediction, have not improved 
our knowledge of the velocity of light, f 

19.10. Velocity of Light in Stationary Matter. The first experiment 
to measure the velocity of light in a transparent substance much denser 

* Valuable summaries of the recent determinations of c, and many original ref- 
erences not given here, will be found in L. Essen, Nature, 165, 583, 1950, and K. D. 
Froome, Proc. Roy. Soc. (London), A213, 123, 1952. 

t The indirect measurements all antedate the determinations in Table 19-1. They 
have been critically reviewed by R. T. Birge, Nature, 134, 771, 1934. 



394 



PHYSICAL OPTICS 



than air was performed in 1850 by Foucault. This was regarded as a 
crucial experiment to decide between the corpuscular and wave theories 
of light. Newton's explanation of refraction by the corpuscular theory 
required that the corpuscles be attracted toward the surface of the denser 
medium, and therefore that they should travel faster in the medium. 
On the wave theory, however, it must be assumed that the light waves 
travel more slowly in the medium. 




Fig. 19G. Foucault's apparatus for determining the velocity of light in water. 

Foucault's apparatus for this experiment is shown in Fig. 19(7. Light 
coming through a slit is reflected from the plane rotating mirror R to the 
equidistant concave mirrors Mi and M 2 . When R is in the position 
(1) the light travels to M\, back along the same path to R, through the 
lens L, and by reflection to the eye at E. When R is in the position (2) 
the light travels the lower path through an auxiliary lens U and tube T 
to M 2 , back to R, through L to G and then to the eye at E. If now the 
tube T is filled with water and the mirror R is set into rotation, there 
will be displacement of the images from E to E x and E 2 . Foucault 
observed that the light ray through the tube was the more displaced. 
This means that it took the light longer to travel the lower path through 
water than it did the upper path through air. The image observed was 
due to a fine wire parallel to and stretched across the slit. Since sharp 
images were desired at Z?i and E if the auxiliary lens U was necessary 
to avoid the effects of refraction at the ends of the tube T. 



THE VELOCITY OF LIGHT 395 

Much more accurate measurements were made by Michelson in 1885. 
Using white light, he found for the ratio of the velocity in air to that in 
water a value of 1.330. A denser medium, carbon disulfide, gave 1.758. 
In the latter case he noticed that the final image of the slit was spread 
out into a short spectrum, which could be explained by the fact that red 
light travels faster than blue light in the medium. The difference in 
velocity between "greenish blue" and "reddish orange" light was 
observed to be 1 or 2 per cent. 

According to the wave theory of light, the index of refraction of a 
medium is equal to the ratio of the velocity of light in vacuum to that 
in the medium. If we compare the above figures with the corresponding 
indices of refraction for white light (water 1.334, carbon disulfide 1.635) 
we find that while the agreement is within the experimental error for 
water, the directly measured value is considerably higher than the 
index of refraction for carbon disulfide. 

This discrepancy is readily explained by the fact that the index of 
refraction represents the ratio of the wave velocities in vacuum and in 
the medium (n = c/v), while the direct measurements give the group 
velocities. Now in a vacuum the two velocities become identical (Sec. 
12.7) and equal to c, so that if we call the group velocity in the medium 
u, the ratios determined by Michelson were values of c/u, rather than 
c/v. The two velocities u and v are related by the general Eq. 12p 

. dv 

u = v ~ x dx 

The variation of v with X may be found by studying the change of the 
index of refraction with color (Sec. 23.2), and it is found that v is greater 
for longer wavelengths, so that dv/d\ is positive. Therefore u should 
be less than v, and this is precisely the result obtained above. Using 
reasonable values for X and dv/dX for white light, the difference between 
the two values for carbon disulfide is in agreement with the theory to 
within the accuracy of the experiments. For water dv/d\ is considerably 
smaller but nevertheless requires that the measured value of c/u should 
be 1.5 per cent higher than c/v. That this is not so indicates an appre- 
ciable error in Michelson's work. The latest work* on the velocity of 
light in water has given agreement not only as to the magnitude of the 
group velocity, but also as to its variation with wavelength. 

At this point it should be emphasized that all the direct methods 
for measuring the velocity of light that we have described give the 
group velocity u and not the wave velocity v. Even though it is not 
evident in the aberration experiment that the wave is divided into 
groups, it should be obvious that since all natural light consists of 

* R. A. Houstoun, Proc. Roy. Soc. Edinburgh, A62, 58, 1944. 



PHYSICAL OPTICS 

wave packets of finite length, any further chopping or modulation 
is immaterial. In air the difference between u and v is small but never- 
theless amounts to 2.2 km/sec. Michelson apparently did not apply 
this correction to his 1926 value, which should therefore have been 
quoted as 299,798 + 4 km/sec. 

19.11. Velocity of Light in Moving Matter. In 1859 Fizeau performed 
an important experiment to determine whether the velocity of light in 
a material medium is affected by motion of the medium relative to the 
source and observer. In Fig. 19// the light from S is split into two 
beams, in much the same way as in the Rayleigh refractometer (Sec. 
13.15). The beams then pass through the tubes A and B containing 



Water 




Fig. 19if . Fizeau's experiment for measuring the velocity of light in a moving medium. 



water flowing rapidly in opposite directions. On reflection from M, the 
beams interchange so that when they reach L\ one has traversed both B 
and A in the same direction as the flowing water while the other has 
traversed A and B in the opposite direction to the flow. The lens L\ 
then brings the beams together to form interference fringes at S'. 

If the light travels more slowly by one route than by the other, 
its optical path has effectively increased and a displacement of the 
fringes should occur. Using tubes 150 cm long and a water velocity of 
700 cm/sec, Fizeau found a shift of 0.46 of a fringe when the direction of 
flow was reversed. This corresponds to an increase in the speed of light 
in one tube, and a decrease in the other, of about half of the velocity 
of the water. 

This experiment was later repeated by Michelson with improved 
apparatus consisting essentially of an adaptation of his interferometer 
to this type of measurement. He observed a shift corresponding to an 
alteration of the velocity of light by 0.434 of the velocity of the water. 

19.12. Fresnel Dragging Coefficient. The above results were com- 
pared with a formula derived by Fresnel in 1818, using the elastic-solid 
theory of the ether. On the assumption that the density of the ether 
in the medium is greater than that in vacuum in the ratio n 2 , he showed 



THE VELOCITY OF LIGHT 



397 



that the ether is effectively dragged along with a moving medium with 
a velocity 



'-O-*) 



(196) 



where v is the velocity of the medium, and n its index of refraction. For 
water, which has n = 1.333 for sodium light, this gives v' = 0.437v, in 
reasonable agreement with Michelson's value 
for white light quoted in the previous para- 
graph. The fraction 1 — (1/n 2 ) will be re- 
ferred to as Fresnel's dragging coefficient. 

19.13. Airy's Experiment. An entirely 
different piece of experimental evidence shows 
that Fresnel's equation must be very nearly 
correct. In 1872 Airy remeasured the angle 
of aberration of light (Sec. 19.2), using a 
telescope filled with water. Upon referring 
to Fig. 1 9.B(&), it will be seen that if the 
velocity of the light with respect to the solar 
system be made less by entering water, one 
would expect the angle of aberration to be 
increased. Actually the most careful meas- 
urements gave the same angle of aberration 
for a telescope filled with water as for one 
filled with air. 

This negative result may be explained by 
assuming that the light is carried along by 
the water in the telescope with the velocity given by Eq. 196. In Fig. 
197, where the angles are of course greatly exaggerated, the velocity now 
becomes c/n and is slightly changed in direction by refraction. If one 
is to observe the ordinary angle of aberration a, it is necessary to add 
to this velocity the extra component v', representing the velocity with 
which the light is dragged by the water. From the geometry of this 
figure, it is possible to prove that v' must obey Eq. 196. The proof will 
not be given here, however, since a different and simpler explanation is 
now accepted, based on the theory of relativity (see Sec. 19.17). 

19.14. Effect of Motion of the Observer. We have seen that in the 
phenomenon of aberration the apparent direction of the light reaching 
the observer is altered when he is in motion. One might therefore 
expect to be able to find an effect of such motion on the magnitude of the 
observed velocity of light. Referring back to Fig. 19B(6), we see that 
the apparent velocity c' = y/sin a is slightly greater than the true 




Fio. 19/. Angle of aberration 
with a water-filled telescope. 



398 



PHYSICAL OPTICS 




Fig. 19J. Velocity of light emitted by a 
moving source. 



velocity c = w/tan a. However, a is a very small ai-gle, so that the 
difference between the sine and the tangent is much smaller than the 
error of measurement of a. A somewhat different experiment embody- 
ing the same principle has been devised, which should be sensitive enough 
to detect this slight change in the apparent velocity if it exists. Before 
describing this experiment, however, we consider in more detail the 

effect of motion of the observer on 
the apparent velocity of light. 

In Fig. 19 J, let the observer at 
be moving toward B with a velocity 
v. Let an instantaneous flash of 
light be sent out at 0. The wave 
will spread out in a circle with its 
center at 0, and after 1 sec the ra- 
dius of this circle will be numeri- 
cally equal to the velocity of light 
c. But during this time the ob- 
server will have moved a distance 
v from to 0'. Hence if the ob- 
server were in some way able to 
follow the progress of the wave, he 
would find an apparent velocity 
which would vary with the direction of observation. In the forward 
direction O'B it would be c - v and in the backward direction O'A, 
c + v. At right angles, in the direction O'P, he would observe a velocity 

Vc 2 - v 2 . 

It is important to notice that in drawing Fig. 19M we have assumed 
that the velocity of the light was not affected by the fact that the source 
was in motion as it emitted the wave. This is to be expected for a 
wave which is set up in a stationary medium, as for instance a sound 
wave in the air. The hypothetical medium carrying light waves is 
the "aether," and if v is the velocity with respect to the aether, the 
same result is expected. For an experiment performed in air, the Fresnel 
dragging coefficient 1 - (1/n 2 ) is so nearly zero that it may be neglected. 
Thus if the observer were moving with the velocity v of the earth in its 
orbit, these considerations lead us to expect the changes in the apparent 
velocity of light described above. Effectively the aether should be mov- 
ing past the earth with a velocity v, and if any effects on the velocity of 
light were found, they could be said to be due to an aether wind or 
aether drift. It would not be surprising if this drift did not correspond 
to the velocity of the earth in its orbit, since we know that the solar 
system as a whole is moving toward the constellation Hercules with a 
velocity of 19 km/sec and it is more reasonable to expect the aether to be 



THE VELOCITY OF LIGHT 



399 



at rest with respect to the system of "fixed stars" than with respect to our 
solar system. 

19.16. The Michelson-Morley Experiment. This experiment, per- 
haps the most famous of any experiment with light, was undertaken in 
1881 to investigate the possible existence of aether drift. In principle 
it consisted merely of observing whether there was any shift of the 
fringes in the Michelson interferom- 
eter when the instrument was turned 
through an angle of 90°. Thus in 
Fig. 19K let us assume that the in- 
terferometer is being carried along 
by the earth in the direction OM*. 
with a velocity v with respect to the 
aether. Let the mirrors Mi and M 2 
be adjusted for parallel light, and 
let OMi = OM 2 = d. The light 
leaving in the forward direction 
will be reflected when the mirror is 
at M'z and will return when the half- 
silvered mirror G has moved to 0" . 
Using the expressions for the veloc- 
ity derived in the previous section, the time required to travel the path 
OM' 2 0" will be 




Fig. 19K. The Michelson interfer- 
ometer as a test for aether drift. 



T: 



d 



c -+- V 



+ 



2cd 



c — V 



and the time to travel OM x O" will be 



To = 



2d 



\/c 2 - v 2 
Each of these expressions may be expanded into series, giving 

onfl m 2d 2d A , v 2 ,Zv* \ 2d/ 1 y 2 \ 

Thus the result of the motion of the interferometer is to increase both 
paths by a slight amount, the increase being twice as large in the direc- 
tion of motion. The difference in time, which would be zero for a sta- 
tionary interferometer, now becomes 






400 PHYSICAL OPTICS 

To change this to path difference we multiply by c, obtaining 



A = d- 2 
pi 



(19c) 



If now the interferometer is turned through 90°, the direction of v is 
unchanged, but the two paths in the interferometer will be interchanged. 




Fig. 19L. Miller's arrangement of the Michelson-Morley experiment to detect aether 
drift. 

This would introduce a path difference A in the opposite sense to that 
obtained before. Hence we expect a shift corresponding to a change of 
path of 2dv 2 /c 2 . 

Michelson and Morley made the distance d large by reflecting the 
light back and forth between 16 mirrors as illustrated in Fig. 19L. 
To avoid distortion of the instrument by strains, it was mounted on 
a large concrete block floating in mercury, and observations were made 
as it was rotated slowly and continuously about a vertical axis. In 
one experiment d was 11 m, so that if we take v = 18.6 mi/sec and 
c = 186.000 mi/sec, we find a change in path of 2.2 X 10~ 5 cm. For 
light of wavelength 6 X 10" 5 , this corresponds to a change of 0.4X, so 
that the fringes should be displaced by two-fifths of a fringe. Careful 



THE VELOCITY OF LIGHT 401 

observations showed that no shift occurred as great as 10 per cent of this 
predicted value. 

This negative result, indicating the absence of an aether drift, was so 
surprising that the experiment has since been repeated with certain 
modifications by a number of different investigators. All have confirmed 
Michelson and Morley in showing that, if a real displacement of the 
fringes exists, it is at most but a small fraction of the expected value. 
The most extensive series of measurements has been made by D. C. 
Miller. His apparatus was essentially that of Michelson and Morley 
(Fig. 19L) but on a larger scale. With a light path of 64 m, Miller 
thought he had obtained evidence for a small shift of about one-thirtieth 
of a fringe, varying periodically with sidereal time. The latest analysis 
of Miller's data, however, makes it probable that the result is not sig- 
nificant, having been caused by slight thermal gradients across the 
interferometer.* 

19.16. Principle of Relativity. The negative result obtained by Michel- 
son and Morley, and by most of those who have repeated their experi- 
ment, forms the basis of the restricted theory of relativity, put forward by 
Einstein t in 1905. The two fundamental postulates on which this 
theory is based are 

1. Principle of Relativity of Uniform Motion. The laws of physics are 
the same for all systems having a uniform motion of translation 
with respect to one another. As a consequence of this, an observer 
in any one system cannot detect the motion of that system by any 
observations confined to the system. 

2. Principle of the Constancy of the Velocity of Light. The velocity of 
light in any given frame reference is independent of the velocity 
of the source. Combined with (1), this means that the velocity of 
light is independent of the relative velocity of the source and 
observer. 

Returning to our illustration (Fig. 19./) of an observer who sends out 
a flash of light at while moving with a velocity v, the above postulates 
would require that any measurements made by the observer at 0' would 
show that he is the center of the spherical wave. But an observer at 
rest at would find that he too is at the center of the wave. The recon- 

* R. S. Shankland, S. W. McCuskey, F. C. Leone, and G. Kuerti, Revs. Mod. Phys., 
27, 167, 1955. 

t Albert Einstein (1879-1955). Formerly director of the Kaiser Wilhelm Institute 
in Berlin, Einstein in 1935 came to the Institute for Advanced Study at Princeton. 
Gifted with one of the most brilliant minds of our times, he has contributed to many 
fields of physics besides relativity. Of prime importance was his famous law of the 
photoelectric effect. He received the N'obel prize in 1921. 



402 PHYSICAL OPTICS 

ciliation of these apparently contradictory statements lies in the fact 
that the space and time scales for the moving system are different from 
those for a fixed system. Events separated in space which are simul- 
taneous to an observer at rest do not appear so to one moving with 
the system. 

The first explanation given for the null result of the Michelson-Morley 
experiment was that the arm of the interferometer that was oriented 
parallel to the earth's motion was decreased in length because of this 
motion. The so-called Fitzgerald-Lorentz contraction required that, if U 
is the length of an object at rest, motion parallel to U with a velocity v 
gives a new length 

l = u{\- j-J) (19d) 

This law would satisfy the condition that the difference in path due to 
aether drift would be just canceled out. Naturally the change in length 
could not be detected by a measuring stick, since the latter would shrink 
in the same proportion. A contraction of this kind should, however, 
bring about changes in other physical properties. Many attempts have 
been made to find evidence for these, but to no avail. According to the 
first postulate of relativity, they must fail. The aether drift does not 
exist, nor is there any contraction for an observer moving along with the 
interferometer. 

Starting from the fundamental postulates of the restricted theory, it is 
possible to show that in a frame of reference that is moving with respect 
to the observer there should actually be changes in the observed values 
of length, mass, and time. The mass of a particle becomes 

m = m a il 2 ) ( 19e ) 

in which mo represents the mass when it is at rest with respect to the 
observer. If light, which has v = c, were to be regarded as consisting 
of particles (see Chap. 30), these would have to have zero rest mass, since 
otherwise m becomes infinite. Experimental measurements have been 
made, mostly with high-speed electrons, which quantitatively verify 
Eq. 19e. Other observable consequences of relativity theory exist, the 
most striking ones being obtained when it is extended to cover accelerated 
systems as well as systems in uniform motion.* From this general theory 
of relativity, predictions are made with regard to the deflection of light 
rays passing close to the sun, and to a decrease in frequency of light emitted 

* For a general account of the theory and its consequences, see R. C. Tolman, 
'•Relativity, Thermodynamics and Cosmology," Oxford University Press, New York, 
1934. Reprinted, 1949. 



THE VELOCITY OF LIGHT 403 

by atoms in a strong gravitational field. Accurate measurements of the 
apparent positions of stars during a total solar eclipse, and of the spectra 
of very dense (white dwarf) stars, have verified these two optical effects. 

These experimental proofs of the theory have been sufficiently con- 
vincing to lead to the general acceptance of the correctness of the theory 
of relativity. While the theory does not directly deny the existence of 
the aether postulated by Fresnel, it says veiy definitely that no experi- 
ment we can ever perform will prove its existence. For if it were possible 
to find the motion of a body with respect to the aether, we could regard 
the aether as a fixed coordinate system with respect to which all motions 
are to be referred. But it is one of the fundamental consequences of 
relativity that any coordinate system is equivalent to any other, and 
no one has any particular claim to finality. Thus, since a fixed aether 
is apparently not observable, there is no reason for retaining the concept. 
It cannot be denied, however, that it is historically important and that 
some of the most important advances in the study of light have come 
through the assumption of a material aether. 

19.17. The Three First-order Relativity Effects. There are three 
optical effects the magnitude of which depends on the first power of 
v/c. They are 

1. The Doppler effect 

2. The aberration of light 

3. The Fresnel dragging coefficient 

Equations for these effects have been derived on the basis of classical 
theory in Sees. 11.6, 19.2, and 19.13. Now it is characteristic of the 
theory of relativity that it yields the same results for first-order effects 
as does the classical theory. Only in second-order effects, which depend 
on v 2 /c 2 , do the predictions of the two theories differ. The Michelson- 
Morley experiment belongs to this class. Even for the first-order effects 
listed above, the results from the two theories differ in the small terms 
of the second and higher power of v/c. In the relativity theory, these 
equations are derived by applying the Lorentz transformation. This is a 
process of translating the description of a motion in terms of one system 
of coordinates into a description of the same motion in terms of another 
system which is in uniform motion with respect to the first. Although 
it is not practicable to give the mathematics of this process here, we shall 
state the chief results and discuss them briefly. 

When the equation for a periodic wave of frequency v is rewritten in the 
coordinates of the observer's frame of reference, the frequency assumes 
a new value given by 



Vl ~ (» 2 A 2 ) „ /, , v 1 »» 1 v* . . \ 



(19/) 



404 PHYSICAL OPTICS 

This is the Doppler effect for the source and observer approaching each 
other with a velocity v along the line joining them. Comparison of the 
series expansion with our previous Eq. HZ shows that the prediction from 
relativity differs from that of the classical theory only in the terms of 
second and higher orders. Theoretically these arise from the fact that 
the rate of a moving clock is slower than that of a stationary one. Ives* 
has given an elegant demonstration of this fact by comparing the fre- 
quency of the radiation emitted by hydrogen atoms in a high-speed beam 
moving first toward the spectroscope, then away from it. In addition 
to the large first-order shifts of the line toward higher and lower fre- 
quencies respectively in these two cases, he observed and measured a small 
additional shift which was toward higher frequencies in both cases. Since 
the term in question contains the square of the velocity, it will be the 
same for either sign of v. This experiment constitutes another verifica- 
tion of the theory of relativity by observation of a second-order effect 
which does not exist according to the classical theory. It might also be 
mentioned that relativity predicts a second-order Doppler shift even 
when the source is moving at right angles to the line of sight. 

The interpretation of the aberration of light and of Airy's experiment 
is simpler from the relativistic point of view. According to the second 
fundamental postulate, the velocity of light must always be c to any 
observer, regardless of his motion. Hence, referring to Fig. 1 92?(&), the 
observed velocity labeled c' must now be labeled c. The formula for 
the angle of aberration, instead of being tan a = v/c, then becomes 

sin a = - (1%) 

c 

It is well known that the sine and the tangent differ only in respect to 
terms of the third and higher orders. Here the angle is so small that in all 
likelihood the difference will never be detected. In Airy's experiment, the 
expectation of observing an increase of the angle when the telescope 
was filled with water arose from the assumption that the water would 
decrease the velocity of the light with respect to the solar system, in 
which the aether was regarded as fixed. But according to the point of 
view of relativity the only "true" velocity of light is its velocity in the 
coordinate system of the observer, and this is inclined at the angle a 
given by Eq. 190. Hence reducing the magnitude of this velocity by 
allowing the fight to enter water will obviously make no change in its 
direction. 

A positive effect corresponding to Fresnel's aether drag can be observed 
when the medium is in motion with respect to the observer (Sec. 19.12), 
but its interpretation by the theory of relativity is entirely different. 
One result of the Lorentz transformation is that two velocities in coordi- 

* H. E. Ives and A. R. Stilwell, J. Opt. Soc. Am., 28, 215, 1938, 31, 369, 1941. 



THE VELOCITY OF LIGHT 405 

nate systems that are in relative motion do not add according the meth- 
ods used in classical mechanics. For example the resultant of two 
velocities in the same line is not their arithmetic sum. Let us call V Q 
the velocity of light in the coordinate system of a moving medium, and 
o the velocity of this medium in the observer's coordinate system. Then 
the resultant velocity V of the light with respect to the observer, instead 
of being merely V + v, must be taken as 

V = 1 + (V /c)(v/c) (19/i) 

The student can easily verify the fact that this equation gives the same 
velocity V for any observer in motion with the velocity v, in the case that 
V = c, that is, in a vacuum. The expression for the Fresnel dragging 
coefficient follows at once from Eq. 19/i, if one neglects second-order 
terms. Thus the binomial expansion gives 



i _X°.»_ 

c c 



) 



= V + v - 



TVv vW 



The last term is again a quantity of the second order and is to be neg- 
lected. Then we obtain, by substituting n for c/Vo, 



n \ n 2 / 



(19i) 



The velocity as seen by the observer is changed by the fraction 1 — (1/n 2 ), 
which is just the value required by Eq. 196. No assumption of any 
"dragging" is involved in the relativity arguments, nor is the existence 
of an aether even postulated. 

PROBLEMS 

1. The innermost satellite of Jupiter has a velocity such that it traverses its own 
diameter in '&-% min. To what fraction of this time would it be necessary to observe the 
instant of an eclipse in order to determine the velocity of light to within ± 100 km /sec? 

2. Assuming the velocity of light to be 299,793 km/sec, and the radius of the earth's 
orbit to be 1.4967 X 10 s km (in computing its velocity), calculate the exact angle of 
aberration according to (a) the classical formula, and (6) the relativity formula. 
Carry through terms of the third order. 

Ans. (a) 20.503 seconds of arc. (6) Relativity gives 1.02 X 10 -6 seconds greater. 

3. At the present time, it is probably more correct to regard the measurements of 
astronomical aberration as determinations of the earth's velocity than of the velocity 
of light. Using the value of the angle of aberration given in Sec. 19.2, and Michel- 
son's 1926 value of c, compute the orbital velocity of the earth. 

4. In Fizeau's toothed-wheel method let L be the distance from the wheel to the 
remote mirror, / the frequency of revolution of the wheel, N the number of teeth, and 



406 PHYSICAL OPTICS 

n the number of the eclipse. Derive an equation giving c in terms of these quantities, 
assuming the adjustment to be made for the minimum of light at the nth eclipse. 

Am, c = LfN/(n-\). 
6. Prove that, in Foucault's rotat ing-mirror arrangement, the intensity of the image 

is proportional to - j — > where u is the distance from the source to the lens, v that from 

the lens to the distant mirror, A the linear aperture of the latter, and r the distance 
from the source to the rotating mirror. 

6. Suppose that 18 m of the distance RM in Foucault's determination (Sec. 19.4) 
were filled with water. Using the group velocities of red and blue light (X = 7200 and 
4000 A) in water, compute the actual length in millimeters of the spectrum he would 
observe. The values of n at these wavelengths are 1.3299 and 1.3432, respectively, 
and those of dn/d\ - 222 and -967 cm -1 . Ans. 0.023 mm. 

7. If the speed of revolution of Michelson's octagonal mirror were exactly 528 rev /sec 
when the image was reflected to its initial position from an adjacent face, find the 
distance to the far mirror. 

8. In the measurement of the velocity of light in the mile-long evacuated pipe by 
Michelson, Pease, and Pearson, a mirror with 32 sides was used. Assuming the total 
path to have been 13 km, and that there was a perfect vacuum in the pipe, use the 
result quoted in Table 19-1 to find the speed of rotation of the mirror required to 
obtain the first undisplaced image. Ans. 720.61 rev /sec. 

9. If Anderson's Kerr-cell apparatus were arranged so that the distance from 
M t to Ma and back (Fig. 19F) comprised 11^ groups, find this distance. The fre- 
quency of his oscillator was 19.2 Mc/sec. 

10. Verify the statement in Sec. 19.11 that a fringe shift of 0.46 fringe in Fizeau's 
experiment corresponds to a change in the velocity of light by about half the velocity 
of water flow. Assuming the effective wavelength and refractive index to be 5500 A 
and 1.333, respectively, find what fraction it actually gives. Ans. 0.508t>. 

11. Assume that, in an experimental measurement of the Fresnel dragging coeffi- 
cient by the interference method, each tube was 2 m long, and the velocity of the water 
6 m/sec. By what fraction of the fringe would the white-light fringe system (X5600) 
be displaced upon reversal of the water stream? 

12. Carbon disulfide has n n = 1.6295 and dn/d\ = -1820 per cm at this wave- 
length. Find (a) the ratio of the velocity of the light in vacuum to the group velocity 
in carbon disulfide, and (6) the exact value of the Fresnel dragging coefficient for this 
substance (see Prob. 14, below). Ans. (a) 1.7367. (6) 0.6892. 

13. Prove from the geometry of Fig. 19/ that, in order for the angle of aberration 
to remain unchanged when the telescope is filled with water, the magnitude of v' must 
be that given by Eq. 196. 

14. Equation 196 needs a small correction arising from the fact that for the mole- 
cules of the moving water the effective frequency is slightly altered by the Doppler 
effect. Prove that this may be taken into account by adding a term —(\/n)(dn/d\) 
to the expression for the dragging coefficient. Here X is the wavelength in vacuum. 
(Hint: Take the refractive index to vary linearly with frequency, and insert the new 
index, as altered by the Doppler effect, in the equation for the velocity of light in the 
moving medium.) 

16. Suppose a Michelson interferometer having arms of length 50 cm is oriented 
so that one arm is parallel to the orbital velocity of the earth. Find the magnitude 
of the Fitzgerald-Lorentz contraction in centimeters of the arm parallel to the earth's 
motion. 

16. Find the mass of an electron which is moving at 2.0 X 10 10 cm /sec. Find also 
the mass of a baseball thrown at 200 ft/sec. The rest masses of the two are 9.106 X 
10-" g and 5^ oz. respectively. Ans. 1.222 X 10~ 27 g. 155.92 -f 3.7 X 10~ 12 g. 



CHAPTER 20 

THE ELECTROMAGNETIC CHARACTER OF LIGHT 



Our study of the properties of light has thus far led us to the conclusion 
that light is a wave motion, propagated with an extremely high velocity. 
In the explanation of interference and diffraction it was not necessary 
to make any assumption as to the nature of the displacement y that 
appears in our wave equations. This is because in these subjects we 
were concerned only with the interaction of light waves with each other. 
In the succeeding chapters we are to consider subjects in which the inter- 
action of light with matter plays a part, and here it becomes necessary 
to specify the physical nature of the quantity y, which is usually termed 
the light vector. Fresnel, who in 1814 first gave the satisfactory explana- 
tion of interference and diffraction by the wave theory, imagined the 
light vector to represent an actual displacement of a material aether, 
which was conceived as an all-pervading substance of very small density 
and of high rigidity. This "elastic-solid" theory had considerable suc- 
cess in interpreting optical phenomena and was strongly supported by 
many leading investigators in the field, such as Lord Kelvin, as late as 
1880. 

20.1. Transverse Nature of Light Vibrations. The principal objection 
to the elastic-solid theory lay in the fact that light had been proved to be 
exclusively a transverse wave motion, i.e., the vibrations are always 
perpendicular to the direction of motion of the waves. No longitudinal 
waves of light have ever been detected. The experimental evidence for 
this comes from the study of the polarization of light (Chap. 24) and is 
perfectly definite, so that we may here take the fact as established. 
Now all elastic solids with which we are familiar are capable of trans- 
mitting longitudinal as well as transverse waves; in fact, under some 
circumstances it is impossible to set up a transverse wave without at the 
same time starting a longitudinal one. Many suggestions were made to 
overcome this difficulty, but all were highly artificial. Furthermore, the 
idea of a material aether itself seemed rather forced, inasmuch as its 
remarkable properties could not be detected by ordinary mechanical 

experiments. 

407 



408 



PHYSICAL OPTICS 



Thus the time was ripe when Maxwell* proposed a theory which not 
only required the vibrations of light to be strictly transverse but also 
gave a definite connection between light and electricity. In a paper read 
before the Royal Society in 1864, entitled "A Dynamical Theory of the 
Electromagnetic Field," Maxwell expressed the results of his theoretical 
investigations in the form of four fundamental equations which have 
since become famous as Maxwell's equations. They were based on the 
earlier experimental researches of Oersted, Faraday, and Joseph Henry 
concerning the relations between electricity and magnetism. They sum- 
marize these relations in concise mathematical form, and constitute a 
starting point for the investigation of all electromagnetic phenomena. 
We shall show in the following sections how they account for the trans- 
verse waves of light. 

20.2. Maxwell's Equations for a Vacuum. The derivation of these 
equations will not be given here, since it would involve a rather extensive 
review of the principles of electricity and magnetism, f Instead we shall 
in this chapter merely state the equations in their simplest form, appli- 
cable to empty space, and then prove that they predict the existence of 
waves having the properties of light waves. The modifications that 
must be introduced in dealing with different kinds of material media will 
be considered at the appropriate places in the following chapters. 

Maxwell's equations may be written as four vector equations, but for 
those unfamiliar with vector notation we shall express them by differen- 
tial equations. In this form the first two equations must be expressed 
by two sets of three equations each. For a vacuum these become, using 
a right-handed set of coordinates, 



(206) 



\dE x 


dH t 


dH y 




ldH x 


dE, 


dEy 


c dt 


dy 


dz 




c dt 


dy 


dz 


IBEy 

c dt 


dH x 
dz 


dH t 

dx 


(20a) 


I dHy 

c dt 


dE x 
dz 


dE t 
dx 


IdE, 


dHy 


dH x 




ldH t 


dEy 


<)/•;, 


c dt 


dx 


dy 




c dt 


dx 


By 



* J. Clerk Maxwell (1831-1879). Professor of experimental physics at Cambridge 
University, England. Contributed a paper to the Royal Society at the age of fifteen. 
Much of his work on the electromagnetic theory was accomplished while an under- 
graduate at Cambridge. His investigations in many fields of physics bear the stamp 
of genius. The kinetic theory of gases was given a solid mathematical foundation by 
Maxwell, whose name is associated with the well-known law of distribution of molec- 
ular velocities. 

t For an elementary derivation, see F. K. Richtmyer and E. H. Kennard, " Introduc- 
tion to Modern Physics," 4th ed., chap. 2, McGraw-Hill Book Company, Inc., New 
York, 1947. 



THE ELECTROMAGNETIC CHARACTER OF LIGHT 409 

The other two equations may be written 

BE* BE, + ^f = (20c) ^£ + 5*1+^-0 (20d) 
dx dy dz dx dy dz 

These partial differential equations give the relations in space and time 
between the vector quantities E, the electric field strength, and H, the 
magnetic field strength. Thus E x , E u , and E„ are the components of E 
along the three rectangular axes, and //*, H y , and H z those of H. The 
electric field is measured in electrostatic units and the magnetic field in 
electromagnetic units. The system which uses electrostatic units for 
all electrical quantities and electromagnetic units for all magnetic ones 
is known as the Gaussian system of units. Although not the most con- 
venient one for practical calculations, it is suitable here, and will always 
be used in what follows. The presence of the important constant c 
in Eqs. 20a and 206 is of course dependent on our choice of units. It 
represents the ratio of the magnitudes of the electromagnetic and electro- 
static units of current. 

Equation 20c merely expresses the fact that no free electric charges 
exist in a vacuum. The impossibility of a free magnetic pole gives rise 
to Eq. 20d. Equations 206 express Faraday's law of induced electro- 
motive force. Thus the quantities occurring on the left side of these 
equations represent the time rate of change of the magnetic field, and 
the spacial distribution of the resulting electric fields occurs on the right 
side. These equations do not give directly the magnitude of the emf, 
but only the rates of change of the electric field along the three axes. 
In particular problems the equations must be integrated to obtain the 
emf itself. 

20.3. Displacement Current. Maxwell's principal new contribution in 
giving these equations was the statement of Eqs. 20a. These come from 
an extension of Ampere's law for the magnetic field due to an electric 
current. The right-hand members give the distribution of the magnetic 
field H in space, but the quantities on the left side do not at first sight 
seem to have anything to do with electric current. They represent the 
time rate of change of the electric field. But Maxwell regarded this 
as the equivalent of a current, the displacement current, which flows as 
long as the electric field is changing and which produces the same mag- 
netic effects as an ordinary conduction current. 

One way of illustrating the equivalence of dE/dt to an electric current 
is shown in Fig. 20 A. Imagine an electric condenser C to be connected 
to a battery B by conducting wires, the whole apparatus being in a 
vacuum with a vacuum between the condenser plates. As the current 



410 



PHYSICAL OPTICS 



i flows for an instant, electric charge accumulates on the plates until the 
condenser is fully charged to the voltage of the battery. Through the 
closed surface S, a certain current has been flowing in during this instant, 
but none has apparently been flowing out. By considerations of con- 
tinuity, Maxwell was led to assume that as much current should flow 
out of such a surface as flows in. But no current of the ordinary sort is 
flowing between the plates of the condenser. The condition of con- 
tinuity can be satisfied only by regarding the change of the electric field 
in this space as the equivalent of a displacement current, the current 
density j of which is proportional to dE/dt. In our system of units this 
current is given by j = l/4ir times dE/dt. It will be noticed that the 

displacement current "flows" in a vac- 
uum, but stops as soon as E becomes 
constant. 

One sees at once the analogy between 
Eqs. 206 and 20a. By Eqs. 206 a chang- 
ing magnetic field produces an emf. 
This was observed by Faraday and is 
very simple to verify experimentally. 
By Eqs. 20a a changing electric field 
should produce a magnetic field ("mag- 
netomotive force ") . This is a much less 
familiar idea and cannot be proved by 
any simple experiment. The reason for 
the difference is that no substance con- 
ducts magnetism as a wire conducts elec- 
tricity. The peculiarity that some sub- 
stances possess of being conductors for 
electricity is the only reason why Eqs. 
206 were discovered before Eqs. 20a. The proof of the correctness of 
Eqs. 20a lies in the remarkable success of Maxwell's equations in account- 
ing for phenomena of nature. It should be noted that Maxwell's equa- 
tions 20a and 206 may be written in terms of the displacement current j 
by replacing the x component (l/c)(dE x /dt) by Ancj 9 and the other com- 
ponents by similar expressions. 

20.4. The Equations for Plane Electromagnetic Waves. Consider the 
case of plane waves traveling in the x direction, so that the wave fronts 
are planes parallel to the y,z plane. If the vibrations are to be repre- 
sented by variations of E and H, we see that in any one wave front they 
must be constant over the whole plane at any instant, and their partial 
derivatives with respect to y and z must be zero. Therefore Eqs. 20a 
to 20d take the form 




Fig. 20A. Illustrating the concept 
of displacement current. 



THE ELECTROMAGNETIC CHARACTER OF LIGHT 



411 



\dE x 
c dt 


= 





1 dE„ 




dH 


c dt 




dx 


1 dE t 




dH, 


c dt 




dx 


dH z 




(t 



--~ =0 



dx 



(20c) 



(20</) 



IdHj 
c dt 
1 dH, 
c dt 
ldH z 
c dt 
dE x 
dx 



dE, 
dx 



dE v 
dx 



= 



(20/) 



(20/i) 



Considering the first equation of Eqs. 20e and Eq. 20h together, it appears 
that the longitudinal component E x is constant in both space and time. 
Similarly from the top line of Eqs. 20/ and from Eq. 20#, H x is also con- 
stant. These components can therefore have nothing to do with the 
wave motion, but must represent constant fields superimposed on the 
system of waves. For the waves themselves, we may therefore write 

E x = H x = 

This means, of course, that the waves are transverse, as stated above. 

Of the four remaining equations, we see that the second equation 20e 
and the third equation 20/ involve E u and H Z) while the third equation 
20e and the second equation 20/ involve E z and H y . Let us assume, for 
example, that E y represents the light vector, so that we are dealing with 
a plane-polarized wave with vibrations in the y direction. We should 



then have to put u t = n u = u, ana consider t 


le two remaining equations 


1 dE v dH, 




c dt dx 

1 dH z dEy 


(20t) 


c dt dx 




We now differentiate the first equation with respect to t and the second 


with respect to x. This gives 


1 d 2 E U d 2 H z 


c dt 2 dxdt 


1 d 2 H t _ d 2 E u 





C dt dx 



dx 2 



Eliminating the derivatives of H z , we find 



d 2 E t 
dt 2 



= c- 



d 2 Ey 

dx 2 



(20j) 



In a similar way, by differentiation of the first equation 20z with respect 
to x and the second with respect to t, we find 



d 2 H z 
dt 2 



= c- 



dx 2 



(20k) 



412 



PHYSICAL OPTICS 



Now Eqs. 20/ and 20k have just the form of the wave equation for 
plane waves (Eq. 116), with E u and H z , respectively, playing the part 
of the displacement y in the two cases. For both, comparison with the 
wave equation shows that the velocity 

v = c (200 

Thus we see that two of the four equations in Eqs. 20e and 20/ predict 
the existence of a wave of the electric vector, plane-polarized in the x,y 
plane, and an accompanying wave of the magnetic vector, plane-polarized 
in the x,z plane. In the form of Eq. 11a they would be represented 
by 

E v = fix ± ct) H z = f(x ± ct) (20m) 

The two waves are interdependent; neither can exist without the other. 
Both are transverse waves, and are propagated in a vacuum with the 
velocity c, the ratio of the electrical units (Sec. 20.2). 

If we had started with the other two equations in Eqs. 20e and 20/, 
we would have obtained another pair of waves, plane-polarized with the 
electric vector in the x,z plane. This pair is quite independent of the 
other, and can exist separately from the other pair. A mixture of the 
two pairs, vibrating at right angles to each other, and with no constant 
phase relation between E v and E z , represents unpolarized light. 

20.5. Pictorial Representation of an Electromagnetic Wave. The 
simplest type of electromagnetic wave is one in which the function / in 
Eq. 20m is a sine or cosine. This is a plane-polarized monochromatic 
plane wave. The three components of E, and the three of H, may for 
such a wave be written 



E x = E v = A sin (ut — kx) 
H x = H u = 



E z = 

H z = A sin {oit — kx) 



(20n) 



By substituting the derivatives of these quantities in Eqs. 20a to 20d, 

it is easily verified that they repre- 
sent a solution of Maxwell's equa- 
tions. 

Figure 20B shows a plot of the 
values of E y and H z along the x 
axis, according to Eq. 20n. In a 
set of plane waves the values of E y 
and H z at any particular value of 
x are the same all over the plane 
x = const.; so this figure merely represents the conditions for one partic- 
ular value of y and z. 

Two important points are to be noticed about Fig. 20B. In the first 




Fig. 205. Distribution of the electric and 
magnetic vectors in a plane-polarized 
monochromatic wave. 



THE ELECTROMAGNETIC CHARACTER OF LIGHT 413 

place, the electric and magnetic components of the wave are in phase 
with each other; i.e., when E y has its maximum value, H z is also a maxi- 
mum. The relative directions of these two vectors, as indicated in the 
figure, agree with Eqs. 20rc. The second point to be noted is that the 
amplitudes of the electric and magnetic vectors are equal. That these 
two are numerically equal in the system of units used here is shown by 
the fact that, in Eqs. 20n, A is the amplitude of each wave. 

20.6. Light Vector in an Electromagnetic Wave. The dual character 
of the electromagnetic wave raises the question as to whether it is the 
electric vector or the magnetic vector which is to be the light vector. 
This question has little meaning, since we could assume either one to 
represent the "displacements" we have been using in previous chapters. 
In every interference or diffraction phenomenon, the electric waves will 
mutually influence each other in exactly the same way as the magnetic 
waves. In one respect, however, the electric component plays a domi- 
nant part. It will be proved in Chap. 25 that it is the electric vector 
that affects the photographic plate and causes fluorescent effects. Pre- 
sumably also the electric vector is the one that affects the retina of the 
eye. In this sense, therefore, the electric wave is the part that really 
constitutes "light," and the magnetic wave, though no less real, is less 
important. 

20.7. Energy and Intensity of an Electromagnetic Wave. The inten- 
sity of mechanical waves was shown in Chap. 11 to be proportional to 
the square of the amplitude. The same result follows from the electro- 
magnetic equations. It can be shown* that in vacuum the electro- 
magnetic field has an energy density given by 

E 2 + H 2 E 2 
Energy per unit volume = — 5 = -j— (20o) 

where E and H are the instantaneous values of the fields, which here are 
equal. Half the energy is associated with the electric vector and half 
with the magnetic vector. The magnitudes of these vectors vary from 
point to point in any wave; so, in order to obtain the energy in any finite 
volume, it is necessary to evaluate the average value of E 2 (or H 2 ). For 
the plane wave of Eq. 20n, one finds that E 2 = \A 2 , the factor -£ being 
the average of the square of the sine over all angles. Hence an electro- 
magnetic wave has an energy density A 2 /8tt, where A is the amplitude 
of either the electric or the magnetic component. 

The intensity of the wave will merely be the product of the above 
expression by the velocity c, since this represents the volume of the wave 

* L. Page and N. I. Adams, Jr., "Principles of Electricity," 2d ed., p. 564, D. Van 
Nostrand Company, Inc., New York, 1949. 



414 PHYSICAL OPTICS 

that will stream through unit area per second. We therefore have 

I - ■£- A* (20p) 

The reader should be reminded that the above statements are applicable 
only to a wave traveling in vacuum. In matter, not only will the velocity 
be different, but also the magnitudes of E and H will no longer be equal. 
Aside from factors of proportionality, however, the intensity is still 
given by the square of the amplitude of either wave (Sec. 23.9). 

20.8. Radiation from an Accelerated Charge. A convenient method 
of representing an electric or magnetic field is by the use of lines of force. 
These are familiar to anyone who has studied elementary electricity and 
magnetism. Each line of force indicates the direction of the field at 
every point along the line, and this is such that a tangent to the line of 
force at any point gives the direction of the force on a small charge or 
pole placed at that point. That is, this tangent gives the direction of 
the electric or magnetic field at that point. 

Consider a small positive electric charge at rest at the point A [Fig. 
20C(a)]. The lines of force are straight lines diverging in every direction 
from the charge and are uniformly distributed in space. The same 
picture would hold if the charge were moving in the direction AB with 
constant velocity, assuming this velocity to be not too large. In these 
two cases — charge at rest and charge in uniform motion — there is no 
radiation of electromagnetic waves. 

In order to produce electromagnetic radiation, it is necessary to have 
acceleration of the charge. A particularly simple case is represented in 
Fig. 20G Y (6). Let the charge, originally at rest at A, be accelerated in 
the direction AC. The acceleration a lasts only until the charge reaches 
the point B, and from that point on the charge moves with a constant 
velocity. In this case we may obtain some information about the form 
of the lines of force radiating from the charge at some later time. Let the 
time of the acceleration from A to B be At, and let the time of the uniform 
motion from B to C be t. When the charge has reached C, at a time 
I + At after it starts, the parts of the original lines of force lying beyond 
the arc RR', drawn about A with the radius c{t + At), cannot have been 
disturbed in any way. This follows from the fact that any electro- 
magnetic disturbance is propagated with the velocity c. At the point C 
the velocity is uniform and the lines of force as far as the arc QQ', drawn 
about B with the radius ct, must be uniform and straight, since the 
charge has had a uniform velocity during the time t. Consequently we 
see that in order to have continuous lines of force they must be con- 
nected through the region between RR' and QQ' somewhat as shown in 
the figure. This gives a pronounced "kink" in each line. The exact 



THE ELECTROMAGNETIC CHARACTER OF LIGHT 



415 



form of the kink will depend upon the type of acceleration existing 
between A and B, that is, whether it is uniform or some type of nonuni- 
form acceleration. 

What is the significance of such a kink in a line of force? If we select 
some point P lying on the kink [Fig. 200(c)], the vector E drawn tangent 
to the line at P gives the actual direction of the field at that point. This 
may be regarded as the resultant of the field E which would be produced 




Fig. 20C. Emission of an electromagnetic pulse from an accelerated charge. 

by the charge at rest, and a transverse field E t . It is the vector E t which 
represents the electric vector of the electromagnetic wave, referred to in 
the foregoing sections. If we carry out this construction for various 
points along the kink, we obtain the variations indicated in Fig. 20C(d) . 
This is obviously not a periodic wave form, but merely a pulse. There 
will be a similar pulse of the H vector at right angles to E t . 

Several important features about the production of electromagnetic 
radiation are illustrated by this example. Most important is the fact 
that E t exists only when the charge is accelerated. No radiation is pro- 
duced if there is no acceleration of charge, and, conversely, an accelerated 
charge will always radiate to a greater or less extent. Also, the example 
shows how the electric field of the radiation can be transverse to the 
direction of propagation. The magnitude of the vector E t obtained by 
the construction of Fig. 20C{d), i.e., the amplitude of the wave, obviously 
depends on the steepness of the kink, and this is determined by how 



416 



PHYSICAL OPTICS 



rapidly the charge was accelerated from A to B. It can be shown theo- 
retically that the rate of radiation of energy from an accelerated charge 
is proportional to the square of the acceleration. Finally, we also find 
that the amplitude of the radiation varies with angle in such a way that 
it is a maximum in directions perpendicular to the line AC and falls to 
zero in both directions along AC. The amplitude is easily shown to be 
proportional to the sine of the angle between AC and the direction 
considered. 

20.9. Radiation from a Charge in Periodic Motion. If the charge in 
Fig. 20C, instead of undergoing a single acceleration, is caused to execute 
a periodic motion, the radiation will be in the form of continuous waves 



Et B 




"W 



^K uiF ^ 




V^v* 1 



(a) (6) 

FlO. 20 D. Emission of electromagnetic waves from a charge in periodic motion. 

instead of a single isolated pulse. Any periodic motion involves accelera- 
tions, and hence will cause the charge to radiate. We shall here consider 
only two especially simple cases, that of linear simple periodic motion 
and that of uniform circular motion. If the positive charge of Fig. 
20D(a) is moved with simple harmonic motion between the limits A and 
B, any line of force will be bent into the form of a sine curve. Let the 
upper curve of Fig. 20D(a) represent one such line, say the one running out 
perpendicular to AB. At the particular instant shown, the electric 
force E at various points along the line has the direction of the tangent 
at those points. Resolving it into the undisturbed field E and the 
transverse component E h we find the various values E t shown just below. 
These also take the form of a sine curve and represent the variation of 
the electric vector along the wave sent out. This is a plane-polarized 
wave. 

In part (6) of the figure, the positive charge is revolving counterclock- 
wise in a circle, in the y,z plane shown in perspective. The same con- 
struction now gives values of E t which are constant in magnitude, but 
vary in direction along the wave. The heads of the arrows he on a 
spiral similar to that of the line of force, but displaced one-quarter of a 
wavelength along the direction of propagation, which here is the x axis. 



THE ELECTROMAGNETIC CHARACTER OF LIGHT 



417 



Source 



To 

induction 
coil 




Spark gap 



Detector 



o 



Fig. 20E. Source and detector of elec- 
tromagnetic waves used by Hertz, 



This screwlike arrangement of the vectors is characteristic of a circularly- 
polarized wave. It is worth pointing out here that, if the radiation 
along the y or z axes were examined, it would be found to be plane- 
polarized in the y,z plane. Actual observation of these two cases is 
possible in the Zeeman effect (Sec. 29.1). 

20.10. Hertz's Verification of the Existence of Electromagnetic Waves. 
We have seen that, starting with a set of equations governing the phe- 
nomena of electromagnetism, Maxwell was able to show the possibility 
of electromagnetic waves and to 
make definite statements about the 
production and properties of the 
waves. Thus he could say that they 
are generated by any accelerated 
charge, that they are transverse waves, 
and that they travel with the velocity 
c in free space. The experimental 
production and detection of the 
waves predicted by Maxwell were 
achieved by Heinrich Hertz.* In 
1887 he began a remarkable series of 
experiments which constitute the 
first important experiments on radio 
waves, i.e., electromagnetic waves of long wavelength. The essential 
features of Hertz's method are illustrated in Fig. 20E. Two plane brass 
plates are connected to a spark gap SG and sparks are caused to jump 
across the gap by charging the plates to high voltage with an induction 
coil. It is known that the discharge of the plates by the spark is an 
oscillatory one. Each time the potential difference between the knobs 
of the gap reaches the point where the air in the gap becomes conducting, 
a spark passes. This represents a sudden surge of electrons across the 
gap, and the signs of the charges on the two plates become reversed. But 
since the air is still conducting, this will produce a return surge, another 
reversal of sign, and the process repeats until the energy is dissipated 
as heat by the resistance of the gap. The frequency of these oscillations 
depends on the inductance and capacity of the circuit. These were very 
small for Hertz's oscillator, and the frequency correspondingly high. In 
some of his experiments it reached 10 9 per sec. Thus we have an electric 
charge undergoing very rapid accelerations, and electromagnetic waves 
should be radiated. 

* Heinrich Hertz (1857-1894). These experiments were carried on while he was 
professor of physics at the Technical High School at Karlsruhe, Germany, in 1885- 
1889. He was then given a professorship at the University of Bonn, which he held 
until his untimely death. 



418 PHYSICAL OPTICS 

In Hertz's experiment the presence of electromagnetic waves was 
detected at some distance from the oscillator by a resonating circuit 
consisting of a circular wire broken by a very narrow spark gap of adjust- 
able length. The changing magnetic field in the wave induced an alter- 
nating emf in the circular wire, whose dimensions were such that the 
natural frequency of its oscillations was the same as that of the source. 
Thus the induced oscillations built up by resonance in the detector until 
they were sufficient to cause sparks to jump the gap. 

It was a simple matter to show that the waves were plane-polarized 
with E in the y direction and H in the z direction. If the loop was 
turned through 90° so that it lay in the x,z plane, the sparks ceased to 
occur. Hertz performed many other experiments with these waves, 
showing among other things that the waves could be reflected and focused 
by curved metal reflectors, and that they could be refracted in passing 
through a large 30° prism of pitch. In these respects they therefore 
showed the same behavior as light waves. 

20.11. Velocity of Electromagnetic Waves in Free Space. The most 
convincing proof of the reality of Hertz's electromagnetic waves lay in 
the demonstration that their velocity was that predicted by the theo- 
retical equation (Eq. 20/). The velocity was measured not directly 
but indirectly by measuring the wavelength. Then from the known 
frequency of the oscillations the velocity could be found by the relation 
v = vk. To measure the wavelength, standing waves were produced by 
interference of the direct waves with those reflected from a plane metal 
reflector. The positions of the nodes could be located by the fact that 
the detector ceased to spark at these points. With a frequency of 
5.5 X 10 7 per sec, X was found to be about 5.4 m, which gives v very close 
to 3 X 10 10 cm/sec. The determination could not be made accurately, 
because the oscillations were highly damped, only three or four occurring 
after each spark, and the wavelength was therefore not accurately defined. 
More recent work by Mercier with undamped waves produced by a 
vacuum-tube oscillator gave the result 2.9978 X 10 10 cm/sec. We have 
already seen, in Sec. 19.8, how the increased precision obtainable with 
cavity resonators has added another significant figure to the velocity 
of light. 

According to Eq. 20/, this observed velocity should equal c, the ratio 
of the emu to the esu of current. As has been mentioned (Sec. 19.9), this 
ratio has been accurately measured by different methods, the most recent 
value being 2.99781 X 10 10 cm/sec. But this is just the measured 
velocity of electromagnetic waves and also agrees exactly with the latest 
measurements of the velocity of light by Michelson and others. For air 
or other gases at atmospheric pressure, a slight modification in the equa- 



THE ELECTROMAGNETIC CHARACTER OF LIGHT 419 

tions is necessary (Chap. 23), but the predicted velocity differs only 
slightly from that in vacuum. 

Hence we are forced to conclude that light consists of electromagnetic 
waves of extremely short wavelength. Beside the evidence of polari- 
zation, which proves that light waves are transverse waves, there is 
much other evidence of this identity. Spectroscopy has shown that the 
atoms contain electrons and that by assuming the acceleration of these 
electrons as they move in orbits around the nucleus one can account for 
the polarization and intensity of the spectrum lines. Furthermore, as 
mentioned in Chap. 11, it has been shown that radio waves, which are 
obviously electromagnetic in character, join continuously onto the region 
of infrared light waves. Thus the explanation of light waves as an 
electromagnetic phenomenon, which in the hands of Maxwell was merely 
a very elegant theory, has since proved to be a reality, and we accept 
the electromagnetic character of light as an established fact. In treat- 
ing the interactions of light with matter we shall therefore use the 
fact that light consists of oscillations of an electric field at right angles 
to the direction of propagation of the waves, accompanied by oscillations 
of the magnetic field, also at right angles to this direction and to the 
direction of the electric field. 

20.12. Cerenkov Radiation. It was stated in Sec. 20.8 that an electric 
charge moving with uniform velocity radiates no energy, but merely 
carries its electromagnetic field along with it. This is true as long as the 
charge is traveling in vacuum. If on the other hand it moves through a 
material medium, as, for example, when a high-speed electron or proton 
enters a piece of glass, it may radiate a small amount of energy even 
though its velocity be constant. The required condition is that the 
speed of the charged particle be greater than the wave velocity c/n of light 
in the medium. It then sets up an impulsive wave similar to the shock 
wave produced by a projectile traveling at a speed greater than that of 
sound. It is of the same character as the "bow wave" of a boat, which 
forms when the boat moves faster than do the water waves. 

The production of this wave is an excellent illustration of the applica- 
tion of Huygens' principle (Sec. 18.1). In Fig. 2QF let e represent an 
electron moving through glass of index 1.50 with a velocity which is 
0.9 of the velocity of light. (To produce such an electron one would 
have to accelerate it through a potential difference of some 650,000 volts.) 
The disturbances produced when the electron occupied successively the 
positions 0, O' , and O" are represented as secondary wavelets which 
have radii OA, O'A', and 0"A", proportional to the elapsed time and 
to their velocity c/n. The resulting wave front is the common tangent 
to these, and takes the form of a cone of half angle 0. Since OA is 



420 



PHYSICAL OPTICS 



normal to the wave front, it will be seen from the figure that 6 is given by 



• „ C 1 
sin 6 = — = — - 
nv n(J 



(20g) 



where v is the velocity of the charged particle and /3 = v/c. If /3 = 0.9 
as in our example, 6 is about 48°. A substantial part of the radiation is 

in visible light, and is detectable by 
the eye or the photographic plate. 
Because of dispersion, the varia- 
tion of n with color, Eq. 20g is not 
perfectly exact. * Furthermore, 
when n is largest (blue light), the 
cone is narrower and the outer edge 
of the conical fan of light rays will 
therefore be blue, while its inner 
edge will be red. 

This type of radiation is now com- 
monly observed with the high-speed 
particles used in nuclear physics. 
By measuring the angle of the cone, 
the velocities and energies of the 
particles may be determined. The 
light resulting from the passage of 
a single particle may be made to register a count with a photomultiplier 
tube. This is the principle of the Cerenkov counter employed by nuclear 
physicists. 




Fig. 20F. Cross section of the conical 
wave produced in Cerenkov radiation. 



PROBLEMS 

1. An oscillator of frequency 35 Mc/sec is set up near a plane metal reflector, and 
the distance between adjacent nodes in the standing waves is found to be 4.28 m. 
Neglecting the refractive index of air, what does this give for the velocity of light? 

2. When a simple harmonic motion is impressed on an electric charge, the lines of 
force at right angles to the motion take the same form as does a stream of water from 
the nozzle of a hose undergoing that motion. The nozzle is continuously pointed 
at right angles to the motion, and gravity is of course neglected. Sketch the form 
of the line after one complete vibration of the source. Remember to add the velocity 
of the hose to that of the water at the middle point of the vibration. 

Ans. Sine wave starting at the nozzle. 

3. Show that Maxwell's equations are satisfied by the solution 

E x = E v = A sin (ut + kz) E, = 

H x = A sin M + kz) H v = H t = 

In which plane is the wave polarized, and in which direction does it travel? 

* For the exact equations, see H. Motz and L. I. Schiff, Am. J. Phys., 21, 258, 1953. 



THE ELECTROMAGNETIC CHARACTER OF LIGHT tZl 

4. Modify Eqs. 20 n so that they represent (a) a plane-polarized wave having 
oscillations of E in the y,z plane, but at 45° to y, and (b) a wave whose oscillations are 
ellipses in the y,z plane (elliptically polarized wave). 

Ans. (a) E x = (b) E» = 

E y = a sin (at — kx) E„ = Oi sin (at — kx) 

E. = a sin (at — kx) E t = a 2 sin (at — kx + 5) 

J7» = tf r = 

ff v = — a sin («/ — kx) H v = — at sin (at — fct 4- 5) 

H t = a sin (a>£ — A;x) #, = oi sin (at — fc:r) 

6. Starting with Eqs. 20n, make a list of the values of all partial derivatives occur- 
ring in Eqs. 20a to 20d. Show by direct substitution that these derivatives satisfy 
the latter equations. 

6. Prove that the segment of the line of force between Q and R in Fig. 20C(b) is a 
straight line when the acceleration of the charge has been uniform. From the slope 
of this segment, show that the ratio Eo/E, falls off as 1/r, and hence that at any 
appreciable distance the transverse component will predominate. 

(Hint: Remember that E is given by Coulomb's law.) 

7. Show that the amplitude of the electromagnetic wave from an accelerated charge 
varies as sin 0, where 6 is the angle between the direction of observation and the direc- 
tion of the acceleration. Make a polar plot of the intensity of the radiation vs. angle. 

8. Show that the ratio of a charge measured in esu to the same charge measured in 
emu has the dimensions of a velocity. (Hint: Start from Coulomb's law in each 
case.) 

9. Calculate the amplitude of the electric field strength in a beam of full sunlight, 
which may be taken as having an intensity of 0.13 watt /cm 1 . 

10. The total force F exerted on a charge e that moves in electric and magnetic 
fields in vacuum is given by 

F = eE + V* 
c 

where it is assumed that the velocity v is perpendicular to the field H . Find the ratio 
of an electric force to the magnetic force exerted on an electron in the first Bohr orbit 
of the hydrogen atom by sunlight which has E - H = 0.0242 (Gaussian units). 

Ans. 137. 

11. Poynting's theorem states that the energy flow in an electromagnetic wave is 
given by 

S = ~ [E X H] 

S is called the Poynting vector, and the expression in square brackets represents the 
vector product. Show that the conclusions of Sees. 20.5 and 20.7 with regard to the 
direction and magnitude of this flow relative to the directions and magnitudes of E and 
H are in agreement with Poynting's theorem. 

12. By assuming Einstein's relation between mass and energy, and taking the mass 
equivalent to an electromagnetic wave to move with the velocity c, derive an expres- 
sion for the pressure that radiation exerts on a perfectly absorbing surface by virtue 
of its momentum. Ans. p = I/c = A 2 /8ir. 

13. A beam of protons of energy 340 Mev is passed through a sheet of extra-dense 
flint glass (n = 1.88). The Cerenkov radiation is found to make an angle of 38° 
with the direction of the proton beam inside the glass. What is the indicated value 
of for these protons? 



CHAPTER 21 

SOURCES OF LIGHT AND THEIR SPECTRA 



Since light is an electromagnetic radiation, we should expect that the 
emission of light from any source results from the acceleration of electric 
charges. It is now certain that the electric charges involved in the 
emission of visible and ultraviolet light are the negative electrons in the 
outer part of the atom. By assuming that vibratory or orbital motions 
of these electrons cause radiation, many of the characteristics of different 
light sources may be explained. It should be emphasized, however, that 
this concept must not be carried too far. In the interpretation of 
spectra it fails in several important respects. These all involve the 
discrete or corpuscular nature of light, which is to be discussed later 
(Chap. 30). For the present, we shall emphasize only those features 
which can be explained by the assumption that light consists of electro- 
magnetic waves. 

21.1. Classification of Sources. Sources of light which are important 
for optical and spectroscopic experiments may be divided into two main 
classes: (1) thermal sources, in which the radiation is the result of high 
temperature, and (2) sources depending on the electrical discharge through 
gases. The sun, with its surface temperature of 5000 to 6000°C, is an 
important example of the first class, but here must also be included such 
important sources as tungsten-filament lamps, the various electric arcs 
at atmospheric pressure, and the flame. Under the second class come 
high-voltage sparks, the glow discharge in vacuum tubes at low pressure, 
and certain low-pressure arcs like the mercury arc. The distinction 
between the two classes is not sharp, and we can go continuously from 
one to the other, for instance by pumping away the air around an electric 
arc. 

21.2. Solids at High Temperature. The majority of practical sources 
for illuminating purposes use the radiation from a hot solid. In the 
tungsten lamp, the filament is heated to about 2100°C by the dissipation of 
electrical energy due to its resistance. The filament can be run at 
temperatures as high as 2300°C but will last for only a short period owing 
to the rapid vaporization of tungsten. In the carbon arc in air, the tem- 
perature of the positive pole is about 4000°C and that of the negative 

422 



SOURCES OF LIGHT AND THEIR SPECTRA 423 

pole, 3000°. The positive pole vaporizes and burns away rather rapidly, 
but it constitutes the brightest thermal source of light available in the 
laboratory. The heating results chiefly from the bombardment of the 
positive pole by electrons drawn from the gaseous part of the arc. Rela- 
tively little light comes from the gas itself. An interesting type of arc, 
useful when a very small source of light is needed, is the so-called con- 
centrated-arc lamp. A simplified diagram of this device is shown in 
Fig. 21.4(a). The cathode consists of a small metal tube packed with 
zirconium oxide, and the anode consists of a metal plate containing a 
hole slightly larger than the end of the cathode. Tungsten, tantalum, 
or molybdenum, because of their high melting points, are used for the 

1 + 



t 



"Nzzzzzz 



a.v:: :::: . 



„ 






to) I (6) 

Fig. 2L4. The concentrated arc, a close approximation to a "point source." 

metal parts. These are sealed into a glass bulb which is filled with an 
inert gas like argon to a pressure of nearly one atmosphere. The arc 
runs between the (fused) surface of the zirconium oxide and the sur- 
rounding anode, as indicated in part (6) of the figure. The tip of the 
cathode is heated by ion bombardment to 2700°C or higher, giving it a 
surface brightness almost equal to that in the carbon arc. The light is 
observed through the hole in the anode, in the direction shown by the 
arrow in Fig. 21 A(a). Lamps of this type can be made in which the 
source is as small as 0.007 cm in diameter. A cheaper way of achieving 
a source of small dimensions is to use a tungsten lamp with a small 
spiral filament (automobile headlight bulb), run at a voltage somewhat 
higher than its rated value. This source does not, however, have the 
smallness and brightness of the concentrated-arc lamp. Other sources 
of continuous spectra will be considered in Sec. 21.9. 

21.3. Metallic Arcs.* When two metal rods connected to a source of 
direct current are touched together and drawn apart, a brilliant arc 
forms between them. A resistance of high current capacity must be 
connected in series with the circuit, and adjusted so that the steady 
current through the arc is from 3 to 5 amp. Higher currents than this 
will cause excessive heating and melting of the electrodes. A large self- 

* These and other sources for use in spectroscopy as well described in G. R. Harrison, 
R. C. Lord, and J. R. Loofbourow, " Practical Spectroscopy," 1st ed., chap. 8, Prentice- 
Hall, Inc., Englewood Cliffs, New Jersey, 1948. 



424 PHYSICAL OPTICS 

inductance in the circuit will stabilize the arc, and a voltage of 220 is 
preferable to 110 in this respect. The two poles are held vertically, in 
line with each other, by clamps with a screw adjustment to vary their 
separation. In the iron arc, the positive pole should be the lower, since 
then a bead of molten iron oxide collects in the small cavity which soon 
forms, and this helps the steadiness of the arc. The radiation from an 
iron, copper, or aluminum arc comes mostly from the gas traversed by 
the arc, this gas consisting almost entirely of the vapor of the metal. 
It has been shown that the gas is at a temperature of from 4000 to 7000°C, 
and it may in cases of very high currents run up to 12,000°C. The 
equivalent of a metallic arc may be obtained with a carbon arc in which 
the positive pole has been bored with an axial hole and packed with the 
salt of a metal, such as calcium fluoride. It is sometimes desirable to 
run a metallic arc in an atmosphere other than air by enclosing it in an 
airtight chamber. The arc may then be run at low pressures as well, 
but this is a difficult procedure. 

With the metals of low melting point, the arc may be permanently 
enclosed in a glass envelope. Of this type are the mercury arc and the 
sodium arc, both commonly used in optical laboratories. In the older 
form of mercury arc, liquid mercury is sealed in a highly evacuated glass 
container of such a shape that the mercury forms two separate pools. 
These make electrical connection with two wires sealed through the glass. 
To start the arc, it is tipped until a thread of mercury connects the two 
pools for an instant and breaks again. As the arc warms up, the pres- 
sure of the mercury vapor increases, and unless a fairly large space is 
available for cooling and condensation, the arc will go out. With suffi- 
cient self-inductance in the circuit, the arc may be run at fairly high 
temperature and pressure, giving a very intense source. For this pur- 
pose the container is made of fused quartz to withstand the higher 
temperature. Quartz has the advantage that it transmits the ultra- 
violet light (Sec. 22.3), and quartz arcs are frequently used in spectros- 
copy and for therapeutic purposes. In using them, great care should 
be taken not to look at the arc too frequently unless glasses are being 
worn, as a painful inflammation of the eyes may result. The same is 
true for the exposed metallic arcs mentioned above. 

As is shown in Fig. 215(a), it is possible to arrange a mercury arc 
to be self -starting. The type illustrated provides an intense, narrow 
vertical source of mercury light suitable for illuminating a slit. The arc 
is formed in a capillary tube of inside diameter 2 mm, and starts a minute 
or so after connecting the terminals to the 110-volt d-c mains. Before 
this time, the current is limited to about 1.5 amp by the resistances 
Ri and R 2 of 80 and 7 ohms, respectively. R 2 is wound on the lower part 
of the capillary and encased in cement so that it heats the mercury at 



SOURCES OF LIGHT AND THEIR SPECTRA 



425 



that point until a bubble of vapor is formed and the mercury thread 
breaks. The resulting arc then generates enough pressure to push the 
mercury above it up to the point A. The arc is then confined to the 
capillary from A to R->. The current has now fallen to about 1.0 amp, 
owing to the additional resistance of the arc itself. 

The sodium arc [Fig. 2\B(b)} is always contained in a double-walled 
envelope made of a special glass that is resistant to blackening by hot 




Sodium 



14cm ((, 




'Anode 



12 volts^- 



+ Filament 
T — *" 'battery 
\- 1.5 volts 



Fig. 215. (a) Small, self-starting mercury arc. (6) Sodium arc. 

sodium vapor. The inner envelope contains argon or neon at low pres- 
sure, and a small amount of metallic sodium. The discharge is initiated 
in the rare gas by electrons emitted from the coiled filament F, and is 
sustained by a relatively small positive potential applied to the anode. 
Since the space between the double walls is highly evacuated to prevent 
heat loss, the interior temperature rises rapidly to the point where the 
sodium melts and vaporizes into the arc. The rare-gas spectrum then 
fades out, being replaced by radiation from the more easily ionized 
atoms of sodium. This is nearly all in the yellow sodium doublet, so 
that the arc yields essentially monochromatic light without the use of 
filters. The doublet is so narrow (separation 5.97 A) that for spectros- 
copy under low dispersion, and for interference measurements with small 
path difference, it may be assumed to be a single line with the average' 
wavelength 5892 A. 

Although they are satisfactory sources for use with small gratings and 



426 PHYSICAL OPTICS 

prism spectroscopes, neither of the above arcs yields spectral lines of 
sufficient sharpness for investigations with very high dispersion. The 
relatively high pressure, temperature, and current density cause a broad- 
ening of the lines, for reasons to be explained in Sec. 21.15. The simplest 
way to produce sharper lines is to use a discharge through a rare gas with 
a small admixture of the metal vapor, and to limit the current to a few 
milliamperes. The discharge may be either a low-voltage arc of the 
type described above or a glow discharge in a vacuum tube (Sec. 21.6). 
Very convenient sources of this type, not only for mercury and sodium, 
but also for cadmium, zinc, and other low-melting metals, may now be 
purchased commercially. In fact, the ordinary mercury fluorescent lamp 
is of the kind required to give sharp lines, and would be satisfactory 
were it not for the coating of fluorescent salt on the inside of the walls. 

21.4. Bunsen Flame. When sufficient air is admitted at the base of a 
bunsen burner, the flame is practically colorless, except for a bluish-green 
cone bounding the inner dark cone of unburnt gas. The temperature 
above the cone is in the neighborhood of 1800°C, high enough to cause 
the emission of light from the salts of certain metals when they are 
introduced into the flame. The color of the flame and its spectrum 
are characteristic of the metal and do not depend on which salt is used. 
The chloride is usually most volatile and gives the most intense coloration. 
The color of the sodium flame is yellow; of strontium, red; of thallium, 
green; etc. For introducing the salt into the flame, a common method 
is to use a loop on the end of a platinum wire, which is first dipped in 
hydrochloric acid and heated until the sodium yellow disappears. Then, 
while red-hot, it is touched to the powdered salt, melting a small amount 
which adheres to the wire. When this is again held in the flame, the 
color is strong but lasts only a short time. A better method is to mix 
a fine spray of the chloride solution with the gas before it enters the 
burner. This is best done with the apparatus shown in Fig. 21C, in 
case air under pressure is available. Air is forced through the atomizer 
S, filling the bottle with a fine spray which is carried into the gas at the 
base of the burner. This gives a very constant light source, and is con- 
venient for the laboratory study of flame spectra. Unfortunately, it can 
be used for only a limited number of metals, the suitable ones including 
lithium, sodium, potassium, rubidium, caesium, magnesium, calcium, 
strontium, barium, zinc, cadmium, indium, and thallium. Other ele- 
ments may be used in the hotter oxygas flame or oxyhydrogen flame, 
but these flames are not as convenient to operate. 

21.6. Spark. By connecting a pair of metal electrodes to the second- 
ary of an induction coil or high-voltage transformer, a series of sparks 
can be made to jump an air gap of several millimeters. If there is no 
capacity in the circuit, the spark is quiet and not very intense, the 



SOURCES OF LIGHT AND THEIR SPECTRA 



427 



radiation coming chiefly from the air in the gap. The spark may be 
made much more violent and brighter by connecting a condenser (sucrt 
as a Leyden jar) in parallel across the gap. We then obtain a condensed 
spark. This is an extremely bright source, the spectrum of which is 
very rich in lines characteristic of the metal of the electrodes. The con- 
densed spark has the drawbacks not only of noisiness and hazard of 
electric shock, but also of the considerable breadth of the lines it emits. 
Nevertheless, it furnishes the most intense excitation available, and ; s the 



Air- 




-Gas 



Fig. 21C. Experimental arrangement for producing spectra by introducing salts of 
metals into the flame of a bunsen burner. 

most efficient source we have for the lines of ionized atoms which have 
lost one or more electrons. Such lines are usually called high-tempera- 
ture, or spark, lines. 

21.6. Vacuum Tube. This is a source that has become increasingly 
common, owing to its application to advertising signs. The familiar red 
"neon signs" contain pure neon gas at a pressure of about 2 cm Hg. 
Metal electrodes are sealed through the ends of the tube, and an electric 
current is caused to traverse the gas by connecting the electrodes to a 
transformer giving a potential of 5000 to 15,000 volts. Other colors are 
produced by introducing a small amount of mercury into a neon or argon 
tube. The heat of the discharge vaporizes the mercury, and we obtain 
the characteristic color and spectrum of mercury vapor. If the tube is 
made of colored glass, certain colors of the mercury light are absorbed 
and various shades of blue and green may be produced. 

In the laboratory, this principle can be used on a smaller scale to 
excite the characteristic radiations of any gas or vapor. Two common 
forms of vacuum tube are illustrated in Fig. 21D. Type (a) is useful 
where maximum intensity is not required, for instance if the tube is to 
be operated with a small induction coil. The electrodes E, E are short 
pieces of aluminum rod, welded to the ends of tungsten wires, the latter 
being sealed through the glass. The light is most intense in the capillary 



428 



PHYSICAL OPTICS 



tube C, where the current density is greatest, and it is observed laterally, 
in the direction indicated by the arrow. Considerably greater intensity 
can be obtained with the "end-on" type shown in (6). Here the elec- 
trodes are of sheet aluminum, rolled up and slipped inside two loosely 
fitting inner glass tubes, G, G. They are fastened to the tungsten leads 
by wrapping a small strip of aluminum at one end around the wire and 
pinching it on tightly. The larger area of the electrodes permits the 



(h) 



15cm 




Fig. 21D. Discharge tubes for obtaining the spectra of gases at low pressure. 

use of greater currents, usually furnished by a transformer, without 
overheating of the electrodes. The light is observed through a plane 
glass window W, which may be fused directly to the tube. The inner 
glass tubes serve to prevent the deposition of aluminum on the outer 
walls of the main tube, which occurs rather rapidly when a tube is used 
at a low pressure. 

The exact pressure at which a vacuum tube should be sealed off varies 
between about 0.5 and 10 mm Hg, according to the gas and to the 
particular spectrum desired. Only a limited number of gases are suit- 
able for long-continued use in a sealed tube of the above type. Of these, 
the rare gases neon, helium, and argon are the most satisfactory. Hydro- 
gen, nitrogen, and carbon dioxide tubes will last only a limited time; the 



SOURCES OF LIGHT AND THEIR SPECTRA 429 

gas gradually disappears from the tube, or "cleans up," until a discharge 
can no longer be maintained. Two processes may be responsible for 
this. The gas may be decomposed by the discharge and the products 
deposited on the walls, or removed by chemical combination with the 
metal electrodes. Or, even with a chemically inert gas, a decrease of 
pressure may be caused by absorption in the above-mentioned metal 
layers that are "sputtered" on the walls from the electrodes. Only the 
main features of the complex phenomena that occur in discharge tubes 
are well understood, and many interesting effects, such as the formation of 
striations, have yet to be explained.* 

21.7. Classification of Spectra. There are two principal classes of 
spectra, known as emission spectra and absorption spectra. In each of 
these there are three types, continuous, line, and band spectra. Emission 
spectra are obtained when the light coming directly from a source is 
examined with a spectroscope. Absorption spectra are obtained when 
the light from a source showing a continuous emission spectrum is passed 
through an absorbing material and thence into the spectroscope. Fig- 
ures 21G, 21 H, and 21J show reproductions of photographed spectra 
illustrating the three types, both in emission and in absorption. Solids 
and liquids, with a few rare exceptions,! give only continuous emission 
and absorption spectra, in which a wide range of wavelengths, without 
any sharp discontinuities, is covered. Discontinuous spectra (line and 
band) are obtained with gases. Gases may also, in certain cases, emit 
or absorb a true continuous spectrum (Sec. 21.9). The three types of 
emission spectra may be easily observed with a carbon arc. If the 
spectroscope is pointed at the white-hot pole of the arc, the spectrum 
is perfectly continuous. If it is pointed at the violet discharge in the 
gas between the poles, bands in the green and violet are seen, and there 
are always a few lines, like the sodium lines, owing to impurities in the 
carbons. 

21.8. Emittance and Absorptance. Although in this chapter we are 
primarily concerned with various sources of light, and hence with emis- 
sion, it will be well to state here a very important relation which exists 
between the emissive and absorptive powers of any surface. A solid, 
when heated, gives a continuous emission spectrum. The amount of 
radiation in this spectrum and its distribution in different wavelengths 

* See L. B. Loeb, "Fundamental Processes of Electrical Discharge in Gases," 
John Wiley & Sons, Inc., New York, 1939. 

t Compounds of some of the rare earth metals give line spectra superposed on a 
continuous spectrum when heated to high temperatures. Their absorption spectra — 
for example, that of didymium glass — show very narrow regions of absorption, which 
at liquid-air temperature become sharp absorption lines. 



430 PHYSICAL OPTICS 

are governed by Kirchhoff's* law of radiation. This states that the 
ratio of the radiant emittance to the absorptance is the same for all bodies 
at a given temperature. As an equation, this law may be written 

— = const. = W B (21a) 

CI 

The quantity W is the total energy radiated per square centimeter of 
surface per second, while a represents the fraction of the incident radia- 
tion which is not reflected or transmitted by the surface. For the 
constant representing this ratio, we have used the symbol Wb, because 
it represents the emittance of a so-called black body. This term specifies 
a body which is perfectly black, i.e., one which absorbs all the radiation 
falling on its surface. Hence for such an ideal body, aii = 1, and Wb 
equals the constant ratio W/a for other bodies. 

Kirchhoff's law expresses a very general relation between the emission 
and absorption of radiation by the surface of different bodies. If the 
absorptance is high, the emittance must also be high. Here it is essen- 
tial to realize the difference between the term absorptance, which measures 
the amount of light disappearing at a single reflection, and the absorption 
within the body of the material, as measured by the absorption coefficient 
a (Sec. 11.5). The latter determines the loss of light upon transmission 
through the material and has no simple connection with the absorptance 
of the surface. In the case of metals, for example, we shall see (Sec. 
25.14) that a very high absorption coefficient is correlated with a high 
reflectance. But a high reflectance also means a low absorptance. Thus 
for metals, and in general for smooth surfaces of pure substances, a high 
absorption coefficient a necessarily means a low absorptance a. 

A black body, which is approximated, for example, by a piece of 
carbon, gives the greatest amount of radiation at a given temperature. 
Transparent or highly reflecting substances are very poor emitters of 
visible light, even when raised to high temperatures. Figure 21 E shows 
a practical illustration of the working of Kirchhoff's law. The right- 
hand picture is a photograph of an ordinary electric iron at room tem- 
perature. A few spots of india ink have been made on the surface, and 
these appear dark since they are regions of high absorptance. The rest 
of the surface is highly reflecting and hence a poor absorber. The 
left-hand photograph was taken by the radiation emitted from the iron 
when heated. The temperature was less than 400°C, so that no visible 
radiation was emitted. However, with infrared-sensitive photographic 
plates a successful photograph was obtained, even though the iron was 

* Gustav Kirchhoff (1824-1887). Professor of physics at Heidelberg and Berlin. 
Beside discovering sonie fundamental laws of electricity, he founded (with Bunsen) 
the science of chemical analysis by spectra. 



SOURCES OF LIGHT AND THEIR SPECTRA 



431 



invisible to the eye in the dark. In this picture, it will be seen that the 
spots which were previously dark (good absorbers) have now become 
brighter than the surroundings, even though they have the same tem- 
perature. Hence they also emit radiation most copiously, as Kirchhoff's 
law requires. Here we are assuming that the ink spots, because they 
are black by visible light, are also good absorbers for infrared light. It 
is in fact essential that W and a refer to the same wavelength, or range 





(a) (6) 

Fig. 212?. Photographs of an electric iron, illustrating Kirchhoff's law of radiation. 
(a) Taken with infrared-sensitive plates, with the iron hot but emitting no visible 
radiation, (b) Taken with ordinary plates and illumination, with the iron at room 
temperature. For the justification of applying the law at different wavelengths, see 
text. (Photographs by H. D. Babcock.) 



of wavelengths, 
we may write 



For the radiation within a small wavelength interval 






(216) 



indicating by the subscript the emittance and absorptance at a particular 
wavelength. This form has important applications to discontinuous 
spectra (Sec. 21.10). 

21.9. Continuous Spectra. The most common sources of continuous 
emission spectra are solids at high temperature,* and some of these 
sources were described in Sec. 21.2. Nothing was said there concerning 
the distribution in different wavelengths of the energy in the continuous 
spectrum. According to Kirchhoff's law, this depends on the ability of 

* A good discussion of the experimental methods employed in this field will be 
found in W. E. Forsythe (ed.), "The Measurement of Radiant Energy," McGraw-Hill 
Book Company, Inc., New York, 1937. 



432 



PHYSICAL OPTICS 



the surface to absorb light of different wavelengths. Thus in a piece of 
china with a red design glazed upon it, the red parts absorb blue and 
violet light more strongly than red. When the piece is heated to a high 
temperature in a furnace and withdrawn, it will be observed that the 
design will appear bluish by the emitted light, since these portions are 
the best absorbers and emitters for blue. In general, therefore, the 
reflectance spectrum of such a solid gives a clue to its emission spectrum. 



0.3r- 




5000 



10,000 



15.000 



20P00A 



-X 



Fig. 21F. Black-body radiation curves plotted to scale. Abscissas give the wave- 
lengths in angstroms and ordinates the energy in calories per square centimeter per 
second in a wavelength interval d\ of 1 A. For numerical values, see "Smithsonian 
Physical Tables," 8th ed., p. 314. 

A black body, which absorbs all wavelengths completely, is commonly 
taken as the standard because it constitutes a particularly simple case 
*vith which the radiation from other substances may be compared. 
Figure 2 IF shows the energy distribution in the radiation from a black 
body at seven different temperatures, and Fig. 21(7 (a) shows photo- 
graphs of the actual spectra corresponding to these curves.* The curve 
for 2000°K represents fairly well that for a tungsten filament, while that 
for 6000°K is closely that of the sun (neglecting the narrow regions of 

* In comparing the spectra of Fig. 2lG(a) with the curves of Fig. 21F it should be 
borne in mind that photographed spectra do not reproduce the true distribution of 
intensity in different wavelengths for three reasons: (1) The dispersion of the prism 
compresses the spectrum at the long-wavelength end. (2) The photographic plate 
is nox equally sensitive to all wavelengths. In particular, the plate used here does not 
respond at all beyond \G600. (3) The blackening of the plate is not proportional to 
the intensity. 



SOURCES OF LIGHT AND THEIR SPECTRA 



433 



absorption due to the Fraunhofer lines). The area under the curve 
represents the total energy emitted in all wavelengths, and increases 
rapidly with the absolute temperature. Calling Wb the total energy in 
ergs emitted from the surface of a black body per square centimeter per 



35 40 45 50 



.4000° C 



(a) 




(b) 



Didymium 
glass 

G Y R 

Fig. 21(7. Continuous spectra, (a) Continuous emission spectra of a solid at the 
three temperatures indicated, taken with a quartz spectrograph. The spectra for 
1000°C and 2000°C were obtained from a tungsten filament. That for 4000°C is 
from the positive pole of a carbon arc. The wavelongth scale is marked in hundreds 
of angstroms, (b) Continuous absorption spectra. The upper spectrum is that of 
the source alone, extending roughly from 4000 to 6500 A. The others show the effect 
on this spectrum of interposing three kinds of colored glass. 

second, and T the absolute temperature, the Stefan-Boltzmann* law 
states that 

W B = <rT* (21c) 

The constant a has the value 5.669 X 10~ 5 erg cm -2 sec -1 °K -4 . The 
wavelength of the maximum of each curve \ m * x depends on the tempera- 
ture according to Wien's^ displacement law, which states that 



UT - const. = 0.2898 cm-deg 



(21d) 



* Ludwig Boltzmann (1844-1906). From 1895 to his death by suicide in 1906, 
professor of physics at Vienna. The law was originally stated by Josef Stefan (1835- 
1893) and was independently demonstrated theoretically by Boltzmann. The latter 
is chiefly known for his contributions to the kinetic theory and the second law of 
thermodynamics. 

t Wilhelm YVien (1861-1928). German physicist, awarded the Nobel prize in 1911 
lor his work in optics and radiation. He also made important discoveries about 
cathode rays and canal rays. 



434 PHYSICAL OPTICS 

where X m « is in centimeters. The shape of the curve itself is given by 
Planck's* law, which may be written 

Wbk d\ = || (e c * Ar - I)" 1 dX (21e) 

Here e is the base of natural logarithms 2.718, while Ci and c 2 are con- 
stants whose values depend on the unit of X. For X in centimeters, 
c, = 3.7413 X 10- 6 erg cm 2 sec" 1 and c 2 = 1.4388 cm-deg. These con- 
stants are of course connected with those in the Stefan-Boltzmann and 
Wien laws, because Eq. 21c can be obtained from Eq. 21e by integrating 
it from X = to X = 00, while Eq. 21d is obtained if we differentiate Eq. 
21e with respect to X and equate to zero to obtain the maximum value. 
Thus, the constant in Eq. 21d is c 2 /4.965. These equations apply, of 
course, only to the radiation from an ideal black body. This can never 
be strictly realized experimentally, but it is approximated by a black 
surface or a hollow cavity with a small opening. The quantity Wb\ dX 
denotes the emission of unpolarized radiation per square centimeter per 
second in all directions in a range d\. 

A source of a continuous spectrum in the ultraviolet region is some- 
times desired for the study of absorption spectra in this region. Hot 
solids are unsuitable for this purpose, because of the relatively small 
amount of ultraviolet light they emit, even at the highest temperatures 
available. It has been found that for this purpose a vacuum-tube dis- 
charge through hydrogen gas at 5 to 10 mm pressure is very satisfactory. 
If a current of a few tenths of an ampere is passed through a tube with 
a rather wide capillary (5 mm diameter) at 2000 volts, a very intense 
continuous spectrum is obtained. The maximum intensity of this con- 
tinuum lies in the violet, but it extends far down into the ultraviolet, to 
about 1700 A. 

21.10. Line Spectra. When the slit of a prism or grating spectroscope 
is illuminated with the light from a mercury arc, several lines of different 
color are seen in the eyepiece. Photographs of common line spectra are 
shown in Fig. 2177(a) to (j). Each of these lines is an image of the slit 
formed by the telescope lens by light of a particular wavelength. The 
different wavelengths are deviated through different angles by the prism 
or grating; hence the line images are separated. It is important to 
realize that line spectra derive their name from the fact that a slit is 
customarily used, whose image constitutes the line. If a point, a disk, 
or any other form of aperture were used in the collimator, the spectrum 
lines would become points, disks, etc., as the case may be. Frequently. 

* Max Planck (1858-1947). Professor at the University of Berlin. He was 
awarded the Nobel prize in 1918 for his derivation of the law of black-body radiation 
and other work in thermodynamics. 






25 



30 35 40 45 50 



liiUillilllilliiillllilllllllliJHI 





Iron 




iliiHliiiiliiiiliiiilniiliiiiliiiiliiiiTiiiiraiiiliiif 




ItfiiiiiiiiiliiiiiiiiiliitiUiWiii ' 

1 1! II 


1 


l| 


Mercury 



ihiiiihTiiiiiiiiteii 



Mercury 



iiliiiiliiiiliiii!iiinmii.nlaJi I il iMiliffliliH'' ' 

I I 



Helium 




Neon 



iiiiliiiiliiiiliiiilniiliiiiliiiiliiiiliiiiii: ' wl'mta&i&Btiikmaatt . 




A 2400' 



(a) 
(b) 

(c) 
(d) 
(e) 
(f) 

(g) 

(h) 
(i) 



Sodium 



A 5850' 



5890 5896 Sun 




Fig. 21 H. Line spectra, (a) Spectrum of the iron arc. The emission spectra (a) to 
(/) were all taken with the same quartz spectrograph. (6) Mercury spectrum from 
an arc enclosed in quartz, (c) Same, from an arc enclosed in glass, (d) Helium in a 
glass discharge tube, (e) Neon in a glass discharge tube. (/) Argon in a glass dis- 
charge tube, (g) Balmer series of hydrogen in the ultraviolet, XX3600 to 4000. This 
is a grating spectrum. The faint lines on either side of the stronger members are false 
lines called "ghosts" (Sec. 17.12). (h) Flash spectrum, showing the emission 
spectrum from the gaseous chromosphere of the sun. This is a grating spectrum 
taken without a slit at the instant immediately preceding a total eclipse, when the 
rest of the sun is covered by the moon's disk. The two strongest images are the H 
and K lines of calcium, and show marked prominences, or clouds of calcium vapor. 
Other strong lines are due to hydrogen and helium, (i) Line absorption spectrum 
of sodium in the ultraviolet, taken with a grating. The bright lines in the background 
arise in the source, which here was a carbon arc. Note the slight continuous absorp- 
tion beyond the series limit. (J) Solar spectrum in the neighborhood of the D lines. 
The two strong lines are absorbed by sodium vapor in the chromosphere, and together 
constitute the first member of the series shown in (i). 

435 



436 



PHYSICAL OPTICS 



in photographing the spectra from astronomical sources, the collimator 
is dispensed with entirely, and a prism or grating placed in front of the 
telescope lens converts the telescope into a spectroscope. In this case, 
each "line" in the spectrum has the shape of the source. For example, 
Fig. 2\H(h) shows the spectrum of the sun at the instant preceding a 
total eclipse, when the usual dark-line absorption spectrum is replaced 
by an emission spectrum from the gases of the solar atmosphere, giving 
the so-called "flash spectrum." The chief use of a slit is to produce 
narrow images, so that the images in different wavelengths do not overlap. 

Table 21-1. Wavelengths, in Angstroms, of Some Useful Spectral Lines 



Sodium 


Mercury 


Helium 


Cadmium 


Hydrogen 


5889.95 s 


4046.56 m 


4387.93 w 


4678.16 m 


6562.82 s 


5895.92 m 


4077.81 in 


4437.55 w 


4799.92 s 


4861.33 m 




4358.35 s 


4471.48 s 


5085.82 s 


4340.46 w 




4916.04 w 


4713.14 m 


6438.47 s 


4101.74 w 




5460.74 s- 


4921.93 m 








5769.59 s 


5015.67 s 








5790.65 s 


5047.74 m; 
5875 . 62 s 
6678.15 m 







The most intense sources of line spectra are metallic arcs and sparks, 
although vacuum tubes containing hydrogen or one of the rare gases are 
very suitable. Flames are often used, because the spectra they give are 
in general simpler, being not so rich in lines. All common sources of 
line emission or line absorption spectra are gases. Furthermore, it is 
now known that only the individual atoms give true line spectra. That 
is, when a molecular compound is used in the source, such as methane 
gas (CH4) in a discharge tube, or sodium chloride in a "cored" carbon 
arc, the lines observed are due to the elements and not to the molecules. 
For example, methane gives a strong line spectrum due to hydrogen, and 
it is well known that sodium chloride gives the yellow sodium lines. 
Lines due to carbon and chlorine do not appear with appreciable intensity 
because these elements are more difficult to excite to emission and their 
strongest lines lie in the ultraviolet and not in the visible part of the 
spectrum. In Table 21-1 are given the wavelengths of the lines in certain 
commonly used emission spectra, with an indication as to whether they 
are strong (s), medium (m), or weak (w). 

Line absorption spectra are obtained only with gases ordinarily com- 
posed of individual atoms (monatomic gases). The absorption lines in 
the solar spectrum are due to atoms which exist as such, rather than 
combined as molecules, only because of the high temperature and low 



SOURCES OF LIGHT AND THEIR SPECTRA 



437 



pressure in the "reversing layer" of the sun's atmosphere [Fig. 21H(h) 
and (J)], In the early days of the study of these lines by Fraunhofer, 
the more prominent ones were designated by letters. The Fraunhofer 
lines are very useful "bench marks" in the spectrum, for instance in the 
measurement and specification of refractive indices. Hence we give 
here, in Table 21-11, their wavelengths and the chemical atoms or mole- 
cules to which they are due. The "lines" A, B, and a are really bands, 
absorbed by the oxygen in the earth's atmosphere. It will be seen 

Table 21-11. The Most Intense Fraunhofer Lines 



Designation 


Origin 


Wavelength, 
k 


Designation 


Origin 


Wavelength, 

A 


A 


o 2 


7594-7621* 


b« 


Mg 


5167.343 


B 


2 


6867-6884* 


c 


Fe 


4957 . 609 


C 


H 


6562.816 


F 


H 


4861.327 


a 


o 2 


6276-6287* 


d 


Fe 


4668.140 


D, 


Na 


5895.923 


e 


Fe 


4383.547 


D, 


Na 


5889.953 


f 


H 


4340.465 


D, 


He 


5875.618 


G 


Fe 


4307.906 


E, 


Fe 


5269.541 


G 


Ca 


4307.741 


b, 


Mg 


5183.618 


g 


Ca 


4226.728 


b, 


Mg 


5172.699 


h 


H 


4101.735 


b, 


Fe 


5168.901 


H 


Ca + 


3968.468 


b« 


Fe 


5167.491 


K 


Ca+ 


3933.666 



Band. 



that b4 and G are blends of two lines which are not ordinarily resolved but 
are due to different elements. 

In the laboratory, there are only a few substances which are suitable 
for observing line absorption spectra, because the absorption lines of 
most monatomic gases lie far in the ultraviolet. The alkali metals are 
one exception, and if sodium is heated in an evacuated steel or pyrex-glass 
tube with glass windows at the ends, the spectrum of light from a tungsten 
source viewed through the tube will show the sodium lines in absorption 
[Fig. 21 1 (i)]. They appear as dark lines against the ordinary continuous 
emission spectrum. 

A somewhat simpler experiment to perform, and one which in addition 
shows the application of Kirchhoff's law to line spectra, is illustrated 
diagrammatically in Fig. 21/. Here A is a horizontal carbon arc cored 
with sodium chloride. The arc is run on a fairly large current so that a 
bright yellow flame F rises above it. If the slit S of the spectroscope is 
directed at the flame, the sodium D lines are seen in emission. They can 
now be observed in absorption by placing a concave mirror M in such a 



438 PHYSICAL OPTICS 

position that it casts an image of the bright positive pole of the arc on 
the slit, the light passing through the flame on its way to the slit. There 
is a considerable concentration of sodium atoms in the flame, and these 
are able to absorb, as well as to emit, the particular frequencies corre- 
sponding to the D lines. Under these circumstances the lines appear 
dark in the spectrum, because of the fact that the flame is at a lower 
temperature than the positive pole. This is a consequence of Kirchhoff's 
law in the form of Eq. 216. To show this, suppose the absorptance a\ 
of the flame for the wavelength of the D lines to be |, so that one-quarter 
of this radiation coming from the mirror is removed from the beam. But 
according to Eq. 216 W\ for this wavelength is l\V B \, that is, the yellow 

8 

L 





Fig. 217. Experimental arrangement for showing absorption of the sodium D lines 
and for illustrating Kirchhoff's law of radiation. 

lines are emitted with one-quarter of the intensity of the corresponding 
portion of the radiation from a black body at the temperature of the flame. 
Hence if the pole of the arc were at the same temperature as the flame, 
the amount absorbed would be just compensated by the emission, and 
no line would appear in the spectrum.* The flame, however, is at a con- 
siderably lower temperature; hence the amount emitted is not enough to 
make up for that absorbed, and dark lines are actually observed with the 
mirror in position. By shifting the mirror so that the image of a cooler 
part of the pole falls on the slit, the lines can be made to disappear, or 
to change into bright lines when the temperature of the selected part of 
the pole is less than that of the flame. 

21.11. Theory of the Connection between Emission and Absorption. 
Kirchhoff's law, as stated in Sec. 21.8, may be proved rigorously by 
thermodynamical methods. However, it will help more in understand- 
ing the above experiment to consider the processes of emission and 
absorption from the electromagnetic standpoint. We may tentatively 
picture! the emission of light as due to periodic motions of the electrons 
in the atoms of the source. These motions would cause electromagnetic 

* We are assuming here that the pole radiates as a perfect black body. 

t That this picture may be only a very approximate one in many instances is 
indicated later in Sec. 21.14. The correspondence principle of quantum theory 
shows, however, that it becomes exact for large orbits (high quantum numbers). 



SOURCES OF LIGHT AND THEIR SPECTRA 439 

waves to be sent out having the same frequencies as the charged particles, 
just as the sound emitted from a tuning fork has the frequency of the 
fork. In the case of sodium vapor, each oscillating charge would be 
regarded as vibrating with a particular frequency, like the tuning fork, 
and the frequency as that of the yellow sodium light. Now if we con- 
sider sodium light to be sent through the vapor, the analogy with the 
tuning fork would still be valid. It is well known that when sound 
waves of the right frequency are incident on a tuning fork, the fork will 
start vibrating and will pick up a considerable amplitude by virtue of 
resonance. In the same way the sodium atoms respond to the incident 
electromagnetic waves, and the energy which they absorb from the waves 
is reemitted as resonance radiation. Although all the energy taken from 
the waves is thus reemitted, resonance radiation is uniformly distributed 
in all directions and thus will be relatively weaker in the forward direction 
than if the absorbing atoms were not present. 

The connection between the emittance and absorptance of a substance 
for light of a given wavelength necessarily follows from the above con- 
siderations. If a substance absorbs light of one frequency strongly, it 
must possess a large number of charges whose characteristic frequencies 
of vibration match that of the light. Conversely, when the substance 
is caused to emit light, these same vibrations will cause strong emission 
of the same frequency. 

21.12. Series of Spectral Lines. In the spectra of some elements, lines 
are observed which obviously belong together to form a series in which 
the spacing and intensities of the lines change in a regular manner. For 
example, in the Balmer series of hydrogen [Fig. 21H(g)} the spacing of the 
lines decreases steadily as they proceed into the ultraviolet toward shorter 
wavelengths, and their intensities fall off rapidly. Although only the 
first four lines he in the visible region, the Balmer series has been traced 
by photography to 31 members in the spectra of hot stars, where it 
appears as a series of absorption lines. The absorption spectrum of 
sodium vapor shows a remarkably long series of lines, each of which is 
a close doublet [not resolved in Fig. 21H(i)], known as the principal 
series. This series also appears in emission from the arc or flame, and 
the well-known D lines constitute the first doublet of the series. Jn 
the sodium spectrum from a flame, about 97 per cent of the intensity 
in this series is in the first member. The emission spectra of the alkalis 
also show two other series of doublets in the visible region, known as 
the sharp and diffuse series. A fourth weak series in the infrared is 
called the fundamental series. The alkaline earth metals, such as cal- 
cium, show two such sets of series — one of single lines, the other of 
triplets. 

A characteristic of any particular series is the approach of the higher 



440 PHYSICAL OPTICS 

series members to a certain limiting wavelength, known as the limit or 
convergence of the series. In approaching this limit, the lines crowd 
closer and closer together, so that there is theoretically an infinite 
number of lines before the limit is actually reached. Beyond the limit 
a rather faint continuous spectrum is sometimes observed in emission; 
in absorption a region of continuous absorption can always be observed 
if the absorbing vapor is sufficiently dense [Fig. 2lII(i)\. The series 
limits furnish the clue to the identification of the type to which the series 
belongs. Thus the sharp and diffuse series approach the same limit, 
while the principal series approaches another limit which for the alkalis 
lies at shorter wavelengths. 

21.13. Band Spectra. The most convenient sources of band spectra 
for laboratory observation are the carbon arc cored with a metallic salt, 
the vacuum tube, and the flame. Calcium or barium salts are suitable 
in the arc or flame, and carbon dioxide or nitrogen in a vacuum tube. 
As observed with a spectroscope of small dispersion, these spectra present 
a typical appearance which distinguishes them at once from line spectra 
[Fig. 21 J (a) to (d)]. Many bands are usually observed, each with a 
sharp edge on one side called the head. From the head, the band shades 
off gradually on the other side. In some band spectra several closely 
adjacent bands, overlapping to form sequences, will be seen [Fig. 21J(6) 
and (d)], while in others the bands are spaced fairly widely, as in Fig. 
21.7(c). When the high dispersion and resolving power of a large grat- 
ing are used, each band is found to be actually composed of many fine 
lines, arranged with obvious regularity into series called branches of the 
band. In Fig. 21J(e), two branches will be seen starting in opposite 
directions from a pronounced gap, where no line appears. In (/) the 
band is double, and the two branches of the left-hand member can be 
seen running side by side. 

Various sorts of evidence point to the conclusion that band spectra 
arise from molecules, i.e., combinations of two or more atoms. Thus it 
is found that, while the atomic or line spectrum of calcium is independent 
of which salt we put in the arc, we obtain different bands by using cal- 
cium fluoride, calcium chloride, or calcium bromide. Also, the bands 
appear in those types of sources where the gas receives less violent 
treatment. Nitrogen in a vacuum tube subjected to an ordinary uncon- 
densed discharge shows only the band spectrum, whereas if a condensed 
discharge is used, the line spectrum appears. The most conclusive evi- 
dence lies in the fact that the absorption spectrum of a gas which is 
known to be molecular (0 2 , N 2 ) shows bands but no hues, owing to the 
absence of any dissociation into atoms. Furthermore, it is found that 
any simple band spectrum, like those described and illustrated above, is 
due to a diatomic molecule. When calcium fluoride (CaF 2 ) is put into 



SOURCES OF LIGHT AND THEIR SPECTRA 



441 



the arc, the bands observed are due to CaF. The violet bands in the 
uncored carbon arc are due to CN, the nitrogen coming from the air 
[Fig. 21.7(e)]. Carbon dioxide in a vacuum tube gives the spectrum of 



40 45 50 




'A3572 Nitric Oxide (NO; 

Fig. 21 J. Band spectra, (a) Spectrum of a discharge tube containing air at low pres- 
sure. Four band systems are present: the y bands of NO (XX2300 to 2700), negative 
nitrogen bands (X 2 +, XX2900 to 3500), second-positive nitrogen bands (N 2 , XX2900 to 
5000), and first-positive nitrogen bands (N a , XX5500 to 7000). (o) Spectrum of a 
high-frequency discharge in lead fluoride vapor. These bands, due to PbF, fall in 
prominent sequences, (c) Spectrum showing part of one band system of SbF, 
obtained by vaporizing antimony fluoride into "active nitrogen." (b) and (c) were 
taken with a large quartz spectrograph, (d) Emission and absorption band spectra 
of BaF. Emission from a carbon arc cored with BaF 2 ; absorption of BaF vapor in an 
evacuated steel furnace. The bands are closely grouped in sequences. Second order 
of 21-ft grating, (e) CN band at X3883 from an argon discharge tube containing 
carbon and nitrogen impurities. Second order of grating. (/) Band in the ultra- 
violet spectrum of NO, obtained from glowing "active nitrogen" containing a small 
amount of oxygen. Second order of grating, (b) and (c) after G. D. Rochester. 

CO, and there are many other examples of this type of dissociation of 
the more complex molecules into diatomic ones. 

21.14. Theory of Line, Band, and Continuous Spectra. The attempt 
to interpret the various definite frequencies emitted by the atoms of a 
gas in producing a line spectrum occupied the best minds in physics dur- 



442 PHYSICAL OPTICS 

ing the early part of this century, and eventually had most important 
consequences. Just as the frequencies of vibration of a violin string 
give sound waves whose frequencies bear the simple ratio of whole 
numbers to the fundamental note, it was first supposed that the fre- 
quencies of the light in the various spectral lines should bear some definite 
relation to each other, which would furnish the clue to the modes of 
vibration of the atom and to its structure. This has proved to be the 
case, though in a very different way than was at first anticipated. The 
definite relation of frequencies is actually found in spectral series. How- 
ever, it will be seen at once that the atomic frequencies do not behave 
like those of a violin string. In the latter the overtones increase steadily 
toward an infinite frequency (zero wavelength), while the frequencies in 
a spectral series approach a definite limiting value. The complete 
explanation of line spectra has now been obtained by developing an 
entirely new theory, called the quantum theory.* Although this theory 
is in many respects in direct contradiction to the electromagnetic theory, 
the latter proved an invaluable guide in attacking such problems as the 
intensity and polarization of spectral lines. It also gave the first clue 
to the behavior of the lines when the source was placed in a magnetic field 
(Chap. 29). For the complete explanation of line spectra, however, the 
quantum theory is absolutely essential. We shall return to this subject 
in the final chapter. 

Band spectra have also required the quantum theory for their complete 
explanation. Nevertheless, the electromagnetic treatment of the prob- 
lem of molecular spectra was somewhat more successful. Certain series 
of bands are observed in the infrared which have frequencies and inten- 
sities related very closely like a fundamental and overtones. These are 
now known to be due to the vibration of the two nuclei in a diatomic 
molecule along the line joining them. The two branches of an individual 
band [Fig. 21/(6)] could be explained as due to rotation of the molecule 
about a direction perpendicular to the above fine. Thus the electro- 
magnetic theory predicts two combination frequencies, the sum and the 
difference of the frequencies of vibration and rotation. This theory, 
however, required a continuous distribution of frequencies in each branch, 
and was unable to explain the discrete lines. 

That a continuous spectrum is obtained from liquids and solids can be 
understood from the fact that here the atoms are closer together than 
in a gas and exert forces on each other. Whereas in a gas the atoms are 
far apart and able to emit definite frequencies, these are so modified by 

* For an elementary treatment of atomic spectra see H. E. White, "Introduction 
to Atomic Spectra," McGraw-Hill Book Company, Inc., New York, 1934. For a dis- 
cussion of band spectra, see G. Herzberg, " Molecular Spectra and Molecular Struc- 
ture. I. Diatomic Molecules," D. Van Nostrand Company, Inc., New York, 1950. 



SOURCES OF LIGHT AND THEIR SPECTRA 443 

the mutual influence of the atoms in a solid that they are spread out 
into a continuous spectrum. The beginning of this effect is seen in the 
spectrum of a gas at a fairly high pressure. The lines become broadened 
because of the more frequent collisions and other influences mentioned 
below. This broadening increases with pressure, so that finally the 
lines merge into a continuous spectrum as the gas approaches the liquid 
state. On the electromagnetic theory, one can understand qualitatively 
the increase in the radiation from a solid with increase of temperature. 
The motions of the charged particles increase in amplitude as the sub- 
stance becomes hotter, with a resultant increase in amplitude of the emit- 
ted waves. More rapid accelerations would cause the average wavelength 
to shift toward higher frequencies as the temperature is raised. Again, 
however, the quantum theory is required to explain the actual distribu- 
tion of energy in different wavelengths. In fact, it was the attempt to 
derive Eq. 21e which first led Planck to make the revolutionary assump- 
tions which constituted the foundations of this theory. 

21.15. Breadth of Spectrum Lines. It was emphasized in Sec. 21.10 
that lines in a spectrum are images of the slit. Hence narrowing the 
latter will sharpen the lines, and the sharpening may continue up to the 
limit set by diffraction (Sec. 15.7). Two causes may, however, prevent 
this theoretical limit from being reached. One of these is most important 
for small spectrographs of low dispersion, and the other for those of very 
high dispersion and resolution. The former cause includes the purely 
geometrical effects such as aberrations of the lenses, imperfections in the 
surfaces or in the homogeneity of the glass prisms, etc. But even if by 
proper design it were possible to eliminate these, and if diffraction were 
negligible, the lines would never approach an infinitesimal width. There 
is still a true, or intrinsic, width of the lines as emitted by the source, 
representing a small spread of wavelengths about the mean position of 
each line. Obviously this will be best revealed by instruments of high 
resolving power, such as a large grating or a Fabry-Perot interferometer. 
It is the cause of the decrease of visibility of the fringes in the Michelson 
interferometer with increasing path difference, which was discussed in 
Sec. 13.12. 

There are three basically different effects contributing to the intrinsic 
line width:* 

1. Shortening of the wave trains. As was indicated in Sec. 12.6, shorter 
trains are equivalent to a greater spread of frequencies. The short- 
ening has two causes: 

* A more quantitative and detailed discussion of line widths will be found in White, 
op. cit., chap. 21. 



444 PHYSICAL OPTICS 

a. Natural damping of the atomic oscillators resulting from the 
radiation of electromagnetic energy. On classical theory, the 
width due to this mechanism is 0.000116 A for a line of any 
wavelength. 

b. Collisions of atoms or molecules, which interrupt the emission of 
continuous waves. 

In the optical region, b is usually much more important than a. 
Since collisions become more frequent as the pressure is raised, the 
broadening from this cause is usually called pressure broadening. 

2. Doppler effect, resulting from the thermal motions of the atoms in 
the light source. Since the velocities are random in direction, and 
widely distributed in magnitude, the frequencies will be shifted 
both up and down by varying amounts. According to the kinetic 
theory, the width due to this cause is proportional to s/T/M, where 
T is the absolute temperature and M the molecular weight. The 
constant of proportionality is 7.16 X 10 -7 X. 

3. Interatomic fields. These may be due to the dipole moments of 
polar molecules, but they are usually the Coulomb fields of the 
ions in a discharge. In Chap. 29 we shall see that spectrum lines are 
split into several components by the action of a uniform electric 
field (Stark effect). Since the interatomic fields are non-uniform in 
both space and time, their effect is to merely broaden the lines. 
Sometimes called Stark broadening, this effect increases rapidly with 
the current density in a discharge. 

Since a broadening due to any of the above causes is mathematically 
equivalent to a more rapid interruption of the wave trains, the separation 
of the second and third effects from the first is justified only by the fact 
that they are observed to vary in the predicted way with the physical 
conditions in the source. 

PROBLEMS 

1. A carbon filament can be run at 2600°C for a short time. Assuming carbon 
to radiate as a black body, find the wavelength at which the most energy is radiated 
from a filament at this temperature. 

2. Find the total power in watts radiated from a metal sphere 2 mm in diameter, 
the sphere being maintained at a temperature of 2000°C. Take the absorptance of 
the surface to be 0.80, and independent of wavelength. Ans. 15.21 watts. 

3. Consider two bodies in an enclosure at a uniform temperature. The nature and 
area of their surfaces need not necessarily be the same, and they may be semitrans- 
parent. From the experimental fact that they come to the same temperature as the 
surroundings, show by considering the energy emitted, absorbed, reflected, and trans- 
mitted by each that Kirchhoff's law of radiation must hold. 

4. A black object becomes barely perceptible to the dark-adapted eye when its 



SOURCES OF LIGHT AND THEIR SPECTRA 445 

temperature reaches 400°C. Find the energy radiated per square centimeter per 
second in a wavelength interval of 10 A at 7200 A under this condition. Find the 
corresponding energy emitted at white heat (1800°C). 

Ans. 2.46 X 10"* erg. 1.26 X 10 6 ergs. 
6. Compare the width due to the Doppler effect of the lines due to helium and 
mercury. Compare also the Doppler widths of either one at 300C° and at the 
temperature of liquid nitrogen ( — 196°C). 

6. A small prism spectrograph has a theoretical resolving power of 5200 at the 
wavelength of the sodium D lines. The prism limits the width of the refracted beam 
to 3.0 cm. The collimator and telescope lenses are each of 30 cm focal length, and the 
slit width is 0.02 mm. Compare the width of one of the D lines due to diffraction, 
due to finite slit width, and due to intrinsic width. For the latter, use the Doppler 
width for a sodium arc at 450°C. Ans. 0.012 mm. 0.020 mm. 0.000123 mm. 

7. From the kinetic-theory equation for the collision frequency in a gas, compute 
the average length of the wave trains emitted by iron vapor at 4000°C and at pressures 
of (a) 1 mm Hg, (b) 760 mm Hg. Using the approximate relation between coherence 
length and line width given in Sec. 1 1 .7, find the corresponding line widths at 5000 A. 
Assume the effective collision diameter of an iron atom to be 2.5 X 10 -8 cm. 



CHAPTER 22 

ABSORPTION AND SCATTERING 



When a beam of light is passed through matter in the solid, liquid, or 
gaseous state, its propagation is affected in two important ways. In the 
first place, the intensity will always decrease to a greater or less extent 
as the light penetrates farther into the medium. In the second place, 
the velocity will be less in the medium than in free space. The loss of 
intensity is chiefly due to absorption, although under some circumstances 
scattering may play an important part. In this chapter we shall discuss 
the consequences of absorption and scattering, while the effect of the 
medium on the velocity, which comes under the term "dispersion," we 
shall consider in the following chapter. The term absorption as used in 
this chapter refers to the decrease of intensity of light as it passes through 
a substance (Sec. 11.5). It is important to distinguish this definition 
from that of absorptance, which was given in Sec. 21.8. The two terms 
refer to different physical quantities, but there are certain relations 
between them, as we shall now see. 

22.1. General and Selective Absorption. A substance is said to show 
general absorption if it reduces the intensity of all wavelengths of light 
by nearly the same amount. For visible light this means that the trans- 
mitted light, as seen by the eye, shows no marked color. There is merely 
a reduction of the total intensity of the white light, and such substances 
therefore appear to be gray. No substance is known which absorbs all 
wavelengths equally, but some, such as suspensions of lamp black or 
thin semitransparent films of platinum, approach this condition over a 
fairly wide range of wavelengths. 

By selective absorption is meant the absorption of certain wavelengths 
of light in preference to others. Practically all colored substances owe 
their color to the existence of selective absorption in some part or parts 
of the visible spectrum. Thus a piece of green glass absorbs completely 
the red and blue ends of the spectrum, the remaining portion in the 
transmitted light giving a resultant sensation of green to the eye. The 
colors of most natural objects such as paints, flowers, etc., are due to 
selective absorption. These objects are said to show pigment or body 
color, as distinguished from surface color, since their color is produced 

446 



ABSORPTION AND SCATTERING 447 

by light which penetrates a certain distance into the substance. Then, 
by scattering or reflection, it is deviated and escapes from the surface, 
but only after it has traversed a certain thickness of the medium and 
has been robbed of the colors which are selectively absorbed. In all 
such cases the absorptance of the body will be proportional to its true 
absorption and will depend in the same way upon wavelength. Surface 
color, on the other hand, has its origin in the process of reflection at the 
surface itself (Sec. 22.7). Some substances, particularly metals like gold 
or copper, have a higher reflecting power for some colors than for others, 
and therefore show color by reflected 
light. The transmitted light here 
has the complementary color, whereas 
in body color the color is the same for 
the transmitted and reflected light. 






mi 



^-'V 

T* 



I 7 - 

A thin gold foil, for example, looks ^ ^ ^^ Qf by findv 

yellow by reflection and blue green by divided particles such as those in smoke, 
transmission. As was mentioned in 

Sec. 21 .8, the body absorption of these materials is very high. This causes 
a high reflectance and a correspondingly low absorptance. 

22.2. Distinction between Absorption and Scattering. In Fig. 22A 
let light of intensity I enter a long glass cylinder filled with smoke. 
The intensity I of the beam emerging from the other end will be less 
than 7 . For a given density of smoke, experiment shows that I depends 
on the length d of the column according to the exponential law stated 
in Sec. 11.5, i.e., 

I = I e- ad (22a) 

Here a is usually called the absorption coefficient, since it is a measure 
of the rate of loss of light from the direct beam. However, most of the 
decrease of intensity of I is in this case not due to a real disappearance 
of the light, but results from the fact that some light is scattered to one 
side by the smoke particles and thus removed from the direct beam. 
Even with a very dilute smoke, a considerable intensity I s of scattered 
light may easily be detected by observing the tube from the side in a 
darkened room. Rays of sunlight seen to cross a room from a window 
are made visible by the fine suspended dust particles present in the air. 
True absorption represents the actual disappearance of the light, the 
energy of which is converted into heat motion of the molecules of the 
absorbing material. This will occur to only a small extent in the above 
experiment, so that the name "absorption coefficient" for a is not appro- 
priate in this case. In general, we can regard a as made up of two 
parts, a a due to true absorption, and a, due to scattering. Equation 22a 
then becomes 

7 = / g-<«-»-«.^ (226) 



448 



PHYSICAL OPTICS 



In many cases either a a or a, may be negligible with respect to the other, 
but it is important to realize the existence of these two different processes 
and the fact that in many cases both may be operating. 

22.3. Absorption by Solids and Liquids. If monochromatic light is 
passed through a certain thickness of a solid or of a liquid enclosed in a 
transparent cell, the intensity of the transmitted light may be much 
smaller than that of the incident light, owing to absorption. If the 
wavelength of the incident light is changed, the amount of absorption 
will also change to a greater or less extent. A simple way of investigating 
the amount of absorption for a wide range of wavelengths simultaneously 



& 



w/;///y///;777z 



WZZZZZmZZZ, 



w 



E 



</////////////%&. 







Fig. 22B. Experimental arrangement for observing the absorption of light by solids, 
liquids, or gases. 

is illustrated in Fig. 22B. S\ is a source which emits a continuous range 
of wavelengths, such as an ordinary tungsten-filament lamp. The light 
from this source is rendered parallel by the lens L\ and traverses a 
certain thickness of the absorbing medium M . It is then focused by L 2 
on the slit <S 2 of a prism spectrograph, and the spectrum photographed 
on the plate P. If M is a "transparent" substance like glass or water, 
the part of the spectrum on P representing visible wavelengths will be 
perfectly continuous, as if M were not present. If M is colored, part 
of the spectrum will be blotted out, corresponding to the wavelengths 
removed by M, and we call this an absorption band. For solids and 
liquids, these bands are almost always continuous in character, fading 
off gradually at the ends. Examples of such absorption bands were 
shown in Fig. 21(7(6). 

Even a substance which is transparent to the visible region will show 
such selective absorption if the observations are extended far enough into 
the infrared or the ultraviolet region. Such an extension involves con- 
siderable experimental difficulty when a prism spectrograph is used, 
because the material of the prism and lenses (usually glass) may itself 
have strong selective absorption in these regions. Thus flint glass cannot 
be used much beyond 25,000 A (or 2.5 n) in the infrared, nor beyond 
about 3800 A in the ultraviolet. Quartz will transmit somewhat farther 



ABSORPTION AND SCATTERING 



440 



in the infrared and much farther in the ultraviolet. Table 22-1 shows 
the limits of the regions over which various transparent substances used 
for prisms will transmit an appreciable amount of light. 

Prisms for infrared investigations are usually of rock salt, while for the 
ultraviolet quartz is most common. In an ultraviolet spectrograph, 
there is no advantage in using fluorite unless air is completely removed 
from the light path, because this begins to absorb strongly below 1850 A. 
Also, specially prepared photographic plates must be used below this 
wavelength, since the gelatin of the emulsion by its absorption renders 

Table 22-1 





Limit of transmission, A 


Substance 


Ultraviolet 


Infrared 




3500 
3800 
1800 
1250 
1750 
1800 
1100 


20.000 




25.000 


Quartz (SiOi) 


40,000 


Fluorite (CaF 2 ) 


95,000 


Rock salt (NaCl) 


145,000 


Sylvin (KC1) 


230,000 




70,000 







ordinary plates insensitive below about 2300 A. In the infrared, photog- 
raphy can now be used as far as 13,000 A, thanks to recently developed 
methods of sensitizing plates. Beyond this, an instrument based upon 
measurement of the heat produced, such as a thermopile, is usually used, 
although as far as 6 n the pholoconductive cell, utilizing the change of 
electrical resistance upon illumination, gives greater sensitivity. 

When absorption measurements are extended over the whole electro- 
magnetic spectrum, it is found that no substance exists which does not 
show strong absorption for some wavelengths. The metals exhibit gen- 
eral absorption, with a very minor dependence on wavelength in most 
cases. There are exceptions to this, however, as in the case of silver, 
which has a pronounced "transmission band" near 3160 A (see Fig. 
25N). A film of silver which is opaque to visible light may be almost 
entirely transparent to ultraviolet light of this wavelength. Dielectric 
materials, which are poor conductors of electricity, exhibit pronounced 
selective absorption which is most easily studied when scattering is 
avoided by having them in a homogeneous condition such as that of a 
single crystal, a liquid, or an amorphous solid. In a general way, it may 
be said that such substances are more or less transparent to X rays and 
7 rays, i.e., light waves of wavelength below about 10 A. Proceeding 
toward longer wavelengths, we encounter a region of very strong absorp- 



450 PHYSICAL OPTICS 

tion in the extreme ultraviolet, which in some cases may extend to the 
visible region, or beyond, and in others may stop somewhere in the near 
ultraviolet (see Table 22-1). In the infrared, further absorption bands 
are encountered, but these eventually give way to almost complete 
transparency in the region of radio waves. Thus for dielectrics we may 
usually expect three large regions of transparency, one at the shortest 
wavelengths, one at intermediate wavelengths (perhaps including the 
visible), and one at very long wavelengths. The limits of these regions 
vary a great deal in different substances, and one substance, such as 
water, may be transparent to the visible but opaque to the near infrared, 
while another, such as rubber, may be opaque to the visible but trans- 
parent to the infrared. 

22.4. Absorption by Gases. The absorption spectra of all gases at 
ordinary pressures show narrow, dark lines. In certain cases it is 
also possible to find regions of continuous absorption (Sec. 21.12), but 
the outstanding characteristic of gaseous spectra is the presence of these 
sharp lines. If the gas is monatomic like helium or mercury vapor, the 
spectrum will be a true line spectrum, frequently showing clearly defined 
series. The number of lines in the absorption spectrum is invariably 
less than in the emission spectrum. For instance, in the case of the 
vapors of the alkali metals, only the lines of the principal series are 
observed under ordinary circumstances [Fig. 21 H(i)]. The absorption 
spectrum is therefore simpler than the emission spectrum. If the gas 
consists of diatomic or polyatomic molecules, the sharp lines form the 
rotational structure of the absorption bands characteristic of mole- 
cules. Here again the absorption spectrum is the simpler, and fewer 
bands are observed in absorption than in emission from the same gas 
[Fig. 2\J{d)]. 

22.5. Resonance and Fluorescence of Gases.* Let us consider what 
happens to the energy of incident light which has been removed by the 
gas. If true absorption exists, according to the definition of Sec. 22.2, 
this energy will all be changed into heat, and the gas will be somewhat 
warmed. Unless the pressure is very low, this is generally the case. 
After an atom or molecule has taken up energy from the light beam, it 
may collide with another particle, and an increase in the average velocity 
of the particles is brought about in such collisions. The length of time 
that an energized atom can exist as such before a collision is only about 
10 -7 or 10 -8 sec, and unless a collision occurs before this time, the atom 
will get rid of its energy as radiation. At low pressures, where the time 
between the collisions is relatively long, the gas will become a secondary 

* A comprehensive discussion of the various aspects of this subject is given in 
A. C. G. Mitchell and M. W. Zemansky, "Resonance Radiation and Excited Atoms," 
The Macmillan Company, New York, 1934. 



ABSORPTION AND SCATTERING 451 

source of radiation, and we do not have true absorption. The reemitted 
light in such cases usually has the same wavelength as the incident light, 
and is then termed resonance radiation (Sec. 21.11). This radiation was 
discovered and extensively investigated by R. W. Wood.* The origin of 
its name is clear, since as has been mentioned the phenomenon is analo- 
gous to the resonance of a tuning fork. Under some circumstances the 
reemitted light may have a longer wavelength than the incident light. 
This effect is called fluorescence. In either resonance or fluorescence, 
some of the light is removed from the direct beam and dark lines will be 
produced in the spectrum of the transmitted light. Resonance and 
fluorescence are not to be classed as scattering. This distinction will 
be made clear in Sec. 22.12. 

Resonance radiation from a gas can readily be demonstrated by the 
use of a sodium-arc lamp. A small lump of metallic sodium is placed 
in a glass bulb connected to a vacuum pump. The sodium is distilled 
from one part of the bulb to another by heating w r ith a bunsen burner, 
thus liberating the large quantities of hydrogen always contained in this 
metal. After a high vacuum is attained, the bulb is sealed off and the 
light of the arc is focused by a lens on the bulb. The bulb must of course 
be observed from the side in a dark room. On gently warming the sodium 
with the flame, a cone of yellow light defining the path of the incident 
light will be seen. At higher temperatures, the glowing cone becomes 
shorter, and eventually is seen merely as a thin bright skin on the inner 
surface of the glass. 

Fluorescence of a gas is most easily shown with iodine vapor, which 
consists of diatomic molecules, I 2 . White light from a carbon arc will 
produce a greenish cone of light when focused in a bulb containing 
iodine vapor in vacuum at room temperature. A still more interesting 
experiment can be performed by using monochromatic light from a 
mercury arc, as shown in Fig. 22C. The source of light is a long hori- 
zontal arc A, which is enclosed in a box with a long slot cut in the top 
parallel to the arc. Immediately above this is a glass tube B filled with 
water. This acts as a cylindrical lens to concentrate the light along the 
axis of tube C, containing the iodine vapor in vacuum. The fluorescent 
light from the vapor is observed with a spectroscope pointed at the plane 
window on the end of tube C. The other end is tapered and painted 
black to prevent reflected light from entering the spectroscope, and a 
screen with a circular hole placed close to the window helps in this 
respect. A polished reflector R laid over C increases the intensity of 

* R. W. Wood (1868-1955). Professor of experimental physics at the Johns Hop- 
kins University. He pioneered in many fields of physical optics and also became one 
of the most colorful figures in American physics. His discoveries in optics are con- 
tained in his excellent text "Physical Optics." 



452 



PHYSICAL OPTICS 



illumination. If B contains a solution of potassium dichromate and 
neodymium sulfate, only the green line of mercury, X5461, is transmitted. 
Figure 22D(b) and (c) were reproduced from a spectrogram taken in this 
way, though with water in the tube B. Beside the lines of the ordinary 
mercury spectrum (marked by dots in the figure) which are present as a 
result of ordinary reflection or Rayleigh scattering (Sec. 22.10), one 
observes a series of almost equally spaced lines extending toward the red 
from the green line. These represent the fluorescent light of modified 
wavelength. 




R 



To 

spectrograph 



m 



5S- 




Fig. 22C. Experimental arrangement for observing the fluorescence of iodine vapor 
with excitation by monochromatic light. 

22.6. Fluorescence of Solids and Liquids. If a solid or a liquid is 
strongly illuminated by light which it is capable of absorbing, it may 
reemit fluorescent light. According to Stokes' law, the wavelength of 
the fluorescent light is always longer than that of the absorbed light. 
A solution of fluorescein in water will absorb the blue portion of white 
light and will fluoresce with light of a greenish hue. Thus a beam of 
white light traversing the solution becomes visible through emission of 
green light when observed from the side but is reddish when looked at 
from the end. Certain solids show a persistence of the reemitted light, 
so that it lasts several seconds or even minutes after the incident light is 
turned off. This is called phosphorescence. 

Very striking fluorescent effects may be produced by illuminating var- 
ious objects with ultraviolet light from a mercury arc. A special nickel 
oxide glass can be obtained which is almost entirely opaque to visible 
light but transmits freely the strong group of mercury lines near X3650. 
If only this light from the arc comes through the glass, many organic as 
well as inorganic substances are rendered visible almost exclusively by 



ABSORPTION AND SCATTERING 



453 



their fluorescent light. The teeth when illuminated by ultraviolet light 
will appear unnaturally bright, but artificial teeth look perfectly black. 

22.7. Selective Reflection. Residual Rays. Substances are said to 
show selective reflection when certain wavelengths are reflected much 
more strongly than others. This usually occurs at those wavelengths 
for which the medium possesses very strong absorption. We are speak- 
ing now of dielectric substances, i.e., those which are nonconductors of 

Fluorescence 




X2536 



Raman Effect 




X4047 



X4358 



X5461 



Fig. 22Z). Photographs of (a) mercury -arc spectrum; (b) fluorescence spectrum of 
iodine; (c) enlarged section of (6); (d) Raman spectrum of hydrogen (after Rasetti); 
(e) Raman spectrum of liquid carbon tetrachloride {after M. Jeppeson); (/) mercury 



electricity. The case of metals is rather different and will be considered 
later in Chap. 25. That there is an intimate connection between selective 
reflection, absorption, and resonance radiation may be seen from an 
interesting observation made by R. W. Wood with mercury vapor. At a 
pressure of a small fraction of a millimeter, mercury vapor shows the 
phenomenon of resonance radiation when illuminated by X2536 from a 
mercury arc. As the pressure of the vapor is increased, the resonance 
radiation becomes more and more concentrated toward the surface of 
the vapor where the incident radiation enters, i.e., on the inner wall of 
the enclosing vessel. Finally, at a sufficiently high pressure, the sec- 
ondary radiation ceases to be visible except when viewed at an angle 
corresponding to the law of reflection. At this angle fully 25 per cent 
of the incident light is reflected in the ordinary way, the remainder having 



454 



PHYSICAL OPTICS 



been absorbed and transformed into heat by atomic collisions. However, 
this high reflection, which is comparable to that of metals in this region, 
exists only for the particular wavelength X2536. Other wavelengths are 
freely transmitted. In this experiment we evidently have a continuous 
transition from resonance radiation to selective reflection. 

A few solids which have strong absorption bands in the visible region 
also show selective reflection. The dye fuchsine is an example. Such 
substances have a peculiar metallic sheen by reflected light and are 
strongly colored. Their color is due to the very high reflection of a 




Fig. 22E. Experimental arrangement for observing residual rays by selective reflection. 



certain band of wavelengths — so high that it is frequently termed "metal- 
lic" reflection. It is this type of reflection that is responsible for surface 
color (Sec. 22.1). 

The most important application of selective reflection has been its use 
in locating absorption bands which lie far in the infrared. For example, 
quartz is found to reflect 80 to 90 per cent of radiation having a wave- 
length of about 8.5 n, or 85,000 A. The method of residual rays for 
isolating a narrow band of wavelengths is based upon this fact.* In 
Fig. 22E, $ is a thermal source of radiation, giving a continuous spectrum. 
After reflection from the four quartz plates Qi to Q\, the radiation is 
analyzed by means of a wire grating G and thermopile T. It is found 
to consist almost entirely of the wavelength 8.5 n. Supposing this 
wavelength to be 90 per cent reflected at each quartz surface, and 
other wavelengths 4 per cent reflected, we have, after four reflections 
(0.9) 4 = 0.66 of the former remaining, but only (0.04) 4 = 0.0000026 of 
the latter. The wavelengths of the residual rays of many substances 
have been measured in this way. Among the longest wavelengths meas- 
ured are those from sodium chloride, potassium chloride, and rubidium 
chloride at 52 n, 63 n, and 74 n, respectively. 

* For more extensive material on this subject, see R. W. Wood, "Physical Optics," 
3d ed., pp. 516-519, The Macmillan Company, New York, 1934. 



ABSORPTION AND SCATTERING 455 

22.8. Theory of the Connection between Absorption and Reflection. 

In Sec. 21.11 we mentioned briefly the mechanism postulated in the 
electromagnetic theory for the production of resonance radiation. It is 
assumed that light waves are incident upon matter which contains 
bound charges capable of vibrating with a natural frequency equal to 
that of the impressed wave. Thus a charge e is acted upon by the 
electric field E with a force eE, and if E varies with a frequency exactly 
matching that with which the charged particle would normally vibrate, 
a large amplitude may be produced. As a result, the charged particle 
will reradiate an electromagnetic wave of the same frequency. In a 
gas at low pressure, where the atoms are relatively far apart, the fre- 
quency which can be absorbed will be sharply defined, and there will 
be no systematic relation between the phases of the light reemitted from 
different particles. The observed intensity from N particles will then 
be just N times that due to one particle (Sec. 12.4). This is the case 
with resonance radiation. 

If, on the other hand, the particles are close together and interacting 
strongly with each other, as in a liquid or solid, the absorption will not 
be limited to a sharply defined frequency but will spread over a con- 
siderable range. The result is that the phases of the reemitted light 
from adjacent particles will agree. This will give rise to regular reflec- 
tion, since the various secondary waves from the atoms in the surface 
will cooperate to produce a reflected wave front traveling off at an angle 
equal to the angle of incidence. In fact, this is just the conception used 
in applying Huy gens' principle to prove the law of reflection. Hence 
selective reflection is also a phenomenon of resonance, and occurs strongly 
near those wavelengths corresponding to natural frequencies of the bound 
charges in the substance. The substance will not transmit light of these 
wavelengths; instead it reflects strongly. True absorption, or the con- 
version of the light energy into heat, may also occur to a greater or less 
extent because of the large amplitudes of the vibrating charges which 
are here involved. If absorption were entirely absent, the reflecting 
power would be 100 per cent at the wavelengths in question. 

22.9. Scattering by Small Particles. The lateral scattering of a beam 
of light as it traverses a cloud of fine suspended matter was mentioned 
in Sec. 22.2. That this phenomenon is closely connected both with 
reflection and with diffraction may be seen by consideration of Fig. 22F. 
In (a) is shown a parallel beam consisting of plane waves advancing 
toward the right and striking a small plane reflecting surface. The suc- 
cessive wave fronts drawn are one wavelength apart, so that here the 
size of the reflector is somewhat greater than a wavelength. The light 
coming off from the surface of the reflector is produced by the vibration 
of the electric charges in the surface with a definite phase relation, and 



456 



PHYSICAL OPTICS 



the spherical wavelets produced by these vibrations cooperate to produce 
short segments of plane wave fronts. These are not sharply bounded 
at their edges by the reflected rays from the edges of the mirror (dotted 
lines) but spread out somewhat, owing to diffraction. In fact, the distri- 
bution of the intensity of the reflected light with angle is just that 
derived in Sec. 15.2 for the light transmitted by a single slit. The 
width of the reflector here takes the place of the slit width, so that we 
shall have greater spreading the smaller the width of the reflector relative 
to the wavelength. 




Fig. 22F. The reflection and diffraction of light by small objects comparable in size 
with the wavelength of light. 

In (6) of the figure, the reflector is much smaller than a wavelength, 
and here the spreading is so great that the reflected waves differ very 
little from uniform spherical waves. In this case the light taken from 
the primary beam is said to be scattered, rather than reflected, since the 
law of reflection has ceased to be applicable. Scattering is therefore a 
special case of diffraction. The wave scattered from an object much 
smaller than a wavelength of light will be spherical, regardless of whether 
or not the object has the plane form assumed in Fig. 22F(b). This fol- 
lows from the fact that there can be no interference between the wavelets 
emitted by the several points on the surface of the scattering particle, 
inasmuch as the extreme points are separated by a distance much less 
than the wavelength. 

The first quantitative study of the laws of scattering by small particles 
was made in 1871 by Rayleigh,* and such scattering is frequently called 

* Several interesting papers laying the foundation of the theory will be found in 
"The Scientific Papers of Lord Rayleigh," vols. 1 and 4, Cambridge I'niversity Press, 
New York, 1912. 



ABSORPTION AND SCATTERING 



457 



Rayleigh scattering. The mathematical investigation of the problem gave 
a general law for the intensity of the scattered light, applicable to any 
particles of index of refraction different from that of the surrounding 
medium. The only restriction is that the linear dimensions of the 
particles be considerably smaller than the wavelength. As we might 
expect, the scattered intensity is found to be proportional to the incident 
intensity and to the square of the volume of the scattering particle. 



120^- 




3000 



4000 



5000 



6000 



7000 



Fig. 22(7. Intensity of scattering vs. wavelength according to Rayleigh's law. 

The most interesting result, however, is the dependence of scattering on 
wavelength. With a given size of the particles, long waves would be 
expected to be less effectively scattered than short ones, because the 
particles present obstructions to the waves which are smaller compared 
with the wavelength for long waves than for short ones. In fact, as will be 
proved in Sec. 22.13, the intensity is proportional to 1/X 4 . Since red light, 
X7200, has a wavelength 1.8 times as great as violet light, X4000, the law 
predicts (1.8) 4 or 10 times greater scattering for the violet light from par- 
ticles much smaller than the wavelength of either color. Figure 22G gives 
a quantitative plot of this relation. 

If white light is scattered from sufficiently fine particles, such as those 
in tobacco smoke, the scattered light always has a bluish color. If the 
size of the particles is increased until they are no longer small compared 
to the wavelength, the light becomes white, as a result of ordinary diffuse 
reflection from the surface of the particles. The blue color seen with 
very small particles, and its dependence on the size of the particles, 



458 PHYSICAL OPTICS 

were first studied experimentally by Tyndall,* and his name is often 
associated with the phenomenon. Chalk dust from an eraser, falling 
across a beam of light from a carbon arc, will illustrate very effectively 
the white light scattered by large particles. 

22.10. Molecular Scattering. Blue Color of the Sky. If a strong 
beam of sunlight is caused to traverse a pure liquid which has been 
carefully prepared to be as free as possible of all suspended particles of 
dust, etc., observation in a dark room will show that there is a small 
amount of bluish light scattered laterally from the beam. Although 
some of this light is still due to microscopic particles in suspension, which 
seem to be almost impossible to eliminate entirely, a certain amount 
appears to be attributable to the scattering by individual molecules of 
the liquid. At first sight it is surprising to find that the scattering from 
liquids is so feeble, in view of the large concentration of molecules present. 
It is, in fact, much weaker than the scattering from the same number of 
molecules of a gas. In the latter, the molecules are randomly distributed 
in space, and in any direction except the forward one the waves scattered 
by different molecules have perfectly random phases. Thus for N mole- 
cules the resultant intensity is just N times that scattered from any 
individual one (see Sec. 12.4). In a liquid, and even more so in a solid, 
the spacial distribution has a certain degree of regularity. Furthermore, 
the forces between molecules act to destroy the independence of phases 
(Sec. 22.8). The result is that the scattering from liquids and solids 
in directions other than forward is very weak indeed. The forward- 
scattered waves are strong and play an essential part in determining the 
velocity of light in the medium, as we shall see in the following chapter. 

Lateral scattering from gases is also weak, but here the weakness is 
due to the relatively smaller number of scattering centers. When a 
great thickness of gas is available, however, as in our atmosphere, the 
scattered light is easily observed. It has been shown by Rayleigh that 
practically all the light that we see in a clear sky is due to scattering 
by the molecules of air. If it were not for our atmosphere, the sky 
would look perfectly black. Actually, molecular scattering causes a con- 
siderable amount of light to reach the observer in directions making an 
angle with that of the direct sunlight, and thus the sky appears bright. 
Its blue color is the result of the greater scattering of short waves. 
Rayleigh measured the relative amount of light of different wavelengths 
in sky light and found rather close agreement with the 1/X 4 law. The 
same phenomenon is responsible for the red color of the sun and sur- 
rounding sky at sunset. In this case, the scattering removes the blue 

*John Tyndall (1820-1893). British "natural philosopher," after 1867 super- 
intendent of the Royal Institution and colleague of Faraday. Tyndall was outstand- 
ing for his ability to popularize and clarify physical discoveries. 



ABSORPTION AN1 SCATTERING 459 

rays from the direct beam more effectively than the red, and the very 
great thickness of the atmosphere traversed gives the transmitted light 
its intense red hue. An experiment demonstrating both the blue of the 
sky and the red of the sun at sunset is described in Sec. 24.15. 

22.11. Raman* Effect. This is a scattering with change of wavelength 
somewhat similar to fluorescence. It differs from fluorescence, however, 
in two important respects. In the first place, the light which is incident 
on the scattering material must have a wavelength that does not corre- 
spond to one of the absorption lines or bands of the material. Otherwise 
we obtain fluorescence, as in the experiment of Sec. 22.5, where the green 
line of mercury is absorbed by the iodine vapor. In the second place, 
the intensity of the light scattered in the Raman effect is much less 
intense than most fluorescent light. For this reason the Raman effect 
is rather difficult to detect, and observations must usually be made by 
photography. 

The apparatus illustrated in Fig. 22C is well adapted to observations 
of the Raman effect, f For this purpose, a liquid or gas which is trans- 
parent to the incident light must be used in the tube C. It is convenient 
to fill tube B with a saturated solution of sodium nitrite, since this 
absorbs the ultraviolet lines of the mercury arc but transmits the blue- 
violet line X4358 with great intensity. Figure 22D{e) shows the Raman 
spectrum of CCU. It will be seen that the same pattern of Raman lines 
is excited by each of the strong mercury lines. Figure 22 D(d) illustrates 
the Raman spectrum of gaseous hydrogen, showing two sets of lines on 
the side toward the red of the exciting line, which in this case was X2536. 
Occasionally still fainter lines are seen on the violet side, two of which 
are visible in (d) and three in (e). This is also sometimes observed in 
the case of fluorescence. Since the modified light in these lines has a 
shorter wavelength than the incident light, they represent a violation of 
Stokes' law (Sec. 22.6) and are called antiStokes lines. 

22.12. Theory of Scattering. When an electromagnetic wave passes 
over a small elastically bound charged particle, the particle will be set 
into motion by the electric field E. In Sec. 22.8 we considered the case 
where the frequency of the wave was equal to the natural frequency of 
free vibration of the particle. We then obtain resonance and fluorescence 
under certain conditions, and selective reflection under others. In both 
cases there may exist a considerable amount of absorption. Scattering, 

* C. V. Raman (1888- ). Professor at the University of Calcutta. He was 
awarded the Nobel prize in 1930 for his work on scattering and for the discovery of 
the effect that bears his name. 

t For a description of the most efficient ways of observing Raman spectra, see 
G. R. Harrison, R. C. Lord, and J. R. Loofbourow, "Practical Spectroscopy," 1st ed., 
Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1948. 



460 PHYSICAL OPTICS 

on the other hand, takes place for frequencies not corresponding to the 
natural frequencies of the particles. The resulting motion of the particles 
is then one of forced vibration. If the particle is bound by a force obey- 
ing Hooke's law, this vibration will have the same frequency and direc- 
tion as that of the electric force in the wave. Its amplitude, however, 
will be very much smaller than that which would be produced by reso- 
nance. Hence the amplitude of the scattered wave will be much less, 
and this accounts for the relative faintness of molecular scattering. The 
phase of the forced vibration will differ from that of the incident wave, 
and this fact is responsible for difference of the velocity of light in the 
medium from that in free space. Thus scattering forms the basis of 
dispersion, which is to be discussed in the following chapter. 

The electromagnetic theory is also capable of giving a qualitative pic- 
ture of the changes of wavelength which occur in the Raman effect and 
in fluorescence. If the charged oscillator is bound by a force which does 
not obey Hooke's law, but some more complicated law, it will be capable 
of reradiating not only the impressed frequency, but also various combi- 
nations of this frequency with the fundamental and overtone frequencies 
of the oscillator. For the complete explanation of these phenomena, 
however, the electromagnetic theory alone is not adequate. It cannot 
explain the actual magnitudes of the changes in frequency nor the fact 
that these are predominantly toward lower frequencies. For this, the 
quantum theory is required. 

Rayleigh scattering yields a characteristic distribution of intensity in 
different directions with respect to that of the primary beam. The scat- 
tered light is also strongly polarized. These features are in general 
agreement with the predictions of the electromagnetic theory. We shall 
not discuss them, however, until we have taken up the subject of polari- 
zation (see Sec. 24.15). 

22.13. Scattering and Refractive Index. The fact that the velocity of 
light in matter differs from that in vacuum is a consequence of scattering. 
The individual molecules scatter a certain part of the light falling on 
them, and the resulting scattered waves interfere with the primary wave, 
bringing about a change of phase which is equivalent to an alteration of 
the wave velocity. This process will be discussed in more detail in the 
chapter which follows, but here some simplified considerations may be 
used to show the connection between scattering and refractive index. 

In Fig. 22H plane waves are shown striking an infinitely wide sheet of 
a transparent material, the thickness of which is small compared to the 
wavelength. Let the electric vector in this incident wave have unit 
amplitude, so that in the exponential notation (Sec. 14.8) it may be repre- 
sented at a particular time by E = e ikx . If the fraction of the wave 
that is scattered is small, the disturbance reaching some point P will be 



ABSORPTION' AND SCATTERING 



461 



essentially the original wave, plus a small contribution due to the light 
scattered by all the atoms in the thin lamina. To evaluate the latter, 
we note that its intensity is pro- 
portional to the coefficient a„ of 
Eq. 226. This measures the frac- 
tional decrease of intensity by scat- 
tering in traversing the small thick- 
ness t, to which the scattered 
intensity must be proportional. 
We therefore have 



-y=a 8 <~7. (22c) 









■r 


^\I2 










^\^ 












Bo 



















Fig. 22H. Geometry of scattering by a 
thin lamina. 



The intensity scattered by a single 

atom, since there are Nt atoms per unit area of the lamina, becomes 



/i 



a,t 

Nt 



a, 

N 



and the amplitude 



These relations hold if the scattered waves from the different centers 
are noncoherent, as is true for the smoke particles discussed in Sec. 22.2. 
The present case of Rayleigh scattering in the forward direction must be 
taken as coherent, however, so that all waves leave the scatterer in phase 
with each other. Then we must add amplitudes instead of intensities, 
and the total scattered amplitude 



K 



**NtJ^ = t V^N 



The complex amplitude at P is obtained by integrating this quantity 
over the surface of the lamina, and adding it to the amplitude of the 
primary wave. The resultant then becomes 

— r? / °° 2trr dr ., „ 



E + E, = e ikR " + t Voc.N 



R 



where the factor 1/R enters because of the inverse-square law. Now 
since R 2 = Rq 2 + r 2 , we have r dr = R dR, and the integral may be 
written 

~ e ikR r dr = 2tt / e ikR dR = — [e ikR ] £ 



Since the wave trains always have a finite length, the scattering as R — * «> 



462" PHYSICAL OPTICS 

can contribute nothing to the coherent wave. Substituting the lower 
limit of the integral, we find 

E + E. = e ikR ' - t V^N * e**' 

i 

= e ikr 

- e ikR °(l + i\t 

By our original assumption, the second term in parentheses is small 
compared with the first. These will be recognized as the first two terms 
in the expansion of e ix ' v ^«^, and may here be equated to it, giving 

E 4- E = e ikIi ''e ixty ^°^ = e i( - kRo+u "^ a '* r > 

Thus the phase of the wave at P has been altered by the amount 
\t -\/a t N. But we know (Sec. 13.15) that the presence of a lamina of thick- 
ness t and refractive index n gives a phase retardation of (2ir/\)(n — l)t. 
Hence 

\t V^N = ^ (n - l)t 

A 

and finally » - 1 - £- V^JV (22d) 

This important relation contains Rayleigh's law of scattering (Sec. 22.9). 
Since, by Eq. 22c, I s is proportional to a„, this scattered intensity varies 
as 1/X 4 , assuming n to be independent of wavelength. In our derivation 
no absorption has been considered, so that the equation is valid only for 
wavelengths well away from any absorption bands. In the next chap- 
ter we shall see how the refractive index behaves as the wavelength 
approaches that of an absorption band. 



PROBLEMS 

1. A certain medium has absorption and scattering coefficients a a and a, of 0.070 
and 0.023 m" 1 , respectively. What fraction of the incident light is transmitted by 
50 m of the medium, and what fraction appears as scattered light? 

2. A tube of smoke 30 cm long transmits 60 per cent of the incident light. After 
precipitation of the smoke particles, it transmits 92 per cent. Calculate the values 
of the absorption and scattering coefficients. Ans. 0.0028 cnr'. 0.0142 cm -1 . 

3. The average lifetime of a sodium atom in the excited state 2 1', from which it emits 
the sodium resonance lines, is 1.6 X 10 -8 sec. When nitrogen is added to sodium 
vapor at low pressure, the resonance radiation is quenched by collisions. If the effec- 
tive collision diameter of a sodium atom with a nitrogen molecule is 7.0 X 10 -8 cm, at 
what pressure does the time between collisions become equal to the above mean 
lifetime? 

4. According to the data given in this chapter, are the residual rays from potassium 
chloride transmitted by rock salt? Ans. No. 



ABSORPTION AND SCATTERING 463 

5. Calculate the ratio of the intensities of Rayleigh scattering for the two mercury 
lines X2536 and X5461. 

6. Photographers know that a yellow filter will "cut" the bluish haze of scattered 
light and give better contrast in a landscape. Assuming the spectral composition 
shown in Fig. 22G, what fraction of the scattered light is removed by a filter that 
absorbs all wavelengths below 4500 A? Transmission of the camera lens and film 
sensitivity limit the normal spectral range of the camera to 3900 to 6000 A. 

Ans. About 49%. 

7. Calculate the lateral dimensions of the two objects illustrated in Fig. 22F, 
assuming that the waves have the value of X appropriate to the green mercury line. 

8. The residual rays after five reflections from a certain crystal are 7 X 10 4 times 
stronger than radiation of adjacent wavelengths. Taking the reflectance at the 
latter wavelengths to be 3.5 per cent, what must be its value at the center of the absorp- 
tion band? Ans. 32.6%. 

9. The common material for green eyeshades looks red when doubled over so that 
one is observing through twice the normal thickness. This effect, known as dichrom-a- 
tism, is due to the presence of two absorption bands with different absorption coeffi- 
cients. Where would these absorption bands have to lie in the above case, and which 
must have the greater coefficient? 

10. Equation 22d is frequently written in terms of the scattering cross section 
a = a,/N, which represents the area of a single atom or molecule that is effective 
in scattering light. Taking the refractive index n D for carbon dioxide under standard 
conditions to be 1.00045, compute the value of a for C0 2 . Ans. 9.18 X 10 -28 cm 1 . 

11. According to Eq. 22d, how should the intensity of the light scattered by a gas 
depend on the pressure of the gas, at constant temperature? Assume the Lorentz- 
Lorenz law (Sec. 13.15) for the dependence of n on density. 

12. The simplest form of dispersion theory, which postulates the existence in each 
atom of a single oscillating charge e of mass m and natural frequency *- , yields 

n _ l= N e>/m 



2ir i> * - 



Assuming Rayleigh's scattering law, find the scattering coefficient a. at X5000 for a 
gas under standard conditions, if the wavelength corresponding to its natural fre- 
quency is 1500 A. Ans. 2.09 X 10~ 8 cm" 1 . 
13. According to the electromagnetic theory, the theoretically significant quantity 
measuring the energy scattered in all directions per unit energy density of the incident 
beam is 8ir« 8 /3. Compute this "scattering coefficient" for helium at 100 atm, given 
that n - 1 is 3.6 X 10~ 3 and X = 5892 A. 



CHAPTER 23 

DISPERSION 



The subject of dispersion concerns the velocity of light in material 
substances, and its variation with wavelength. Since the velocity is 
c/n, any change in refractive index n entails a corresponding change of 
velocity. We have seen in Sec. 1.7 that the dispersion of color which 
occurs upon refraction at a boundary between two different substances is 
direct evidence of the dependence of the n's on wavelength. In fact, 
measurements of the deviations of several spectral lines by a prism furnish 

the most accurate means of de- 
termining the refractive index, 
and hence the velocity, as a func- 
tion of wavelength. 

23.1. Dispersion of a Prism. 
When a ray traverses a prism, as 
shown in Fig. 23 A, we can measure 
with a spectrometer the angles of 
emergence 6 of the various wave- 




Fig. 23A. Refraction by a prism at mini- 
mum deviation. 



lengths. The rate of change, dd/d\, is called the angular dispersion of the 
prism. It can be conveniently represented as the product of two factors, 
by writing 



dO 
dX 



dd dn 
dn dX 



(23a) 



The first factor may be evaluated by geometrical considerations alone, 
while the second is a characteristic property of the prism material, usually 
referred to simply as its dispersion. Before considering the latter quan- 
tity, let us evaluate the geometrical factor dO/dn for a prism, in the 
special case of minimum deviation. 

For a given angle of incidence on the second face of the prism, we may 
differentiate Snell's law of refraction n = sin 0/sin 0, regarding sin <£ as 
a constant. We obtain 

d6 _ sin <j> 

dn cos 

This is not, however, the value to be used in Eq. 23a, which requires 

464 



DISPERSION 



465 



the rate of change of for a fixed direction of the rays incident on the first 
face. Because of the symmetry in the case of minimum deviation, it is 
obvious that equal deviations occur at the two faces, so that the total 
rate of change of will be just twice the above value. We then have 

d0 = 2 sin _ 2 sin (a/2) 
dn cos 9 cos 6 



where a is the refracting angle of the prism. The result becomes still 
simpler when expressed in terms of lengths rather than angles. Designat- 
ing by s, B, and b the lengths shown in Fig. 23 A, we may write 



dd 
dn 



2s sin (a/2) 
s cos 6 



b 



(236) 



Hence the required geometrical factor is just the ratio of the base of the 
prism to the linear aperture of the emergent beam, a quantity not far 
different from unity. The angular dispersion becomes 



de 



Bdn 
b d\ 



(23c) 



In connection with this equation, it is to be noted that the equation 
for the chromatic resolving power (Eq. 15h) follows very simply from it 
upon the substitution of X/6 for dd. 

23.2. Normal Dispersion. In considering the second factor in Eq. 23a, 
let us start by reviewing some of the known facts about the variation of n 
with X. Measurements for some typical kinds of glass give the results 
shown in Table 23-1. If any set of values of n is plotted against wave- 

Table 23-1. Refractive Indices and Dispersions for Several Common Types of 
Optical Glass (Unit of Dispersion, 1/A) 





Telescope crown 


Borosilicatc crown 


Barium flint 


Vitreous quartz 


Wave- 










length, 


















X, in A. 




dn 




dn 




dn 




dn 






d\ 




d\ 


n 


d\ 


n 


<*X 


C 6563 


1.52441 


0.35 X 10"» 


1 . 50883 


0.31 X 10-* 


1 . 58848 


0.38 X 10-* 


1.45640 


0.27 X 10"» 


6439 


1.52490 


0.36 X 10"* 


1.50917 


0.32 X 10-' 


1.58896| 0.39 X 10"' 


1.45674 


0.28 X 10-' 


D 5890 


1 . 52704 


0.43 X 10~ s 


1.51124 


0.41 X 10-* 


1.59144 0.50 X 10-' 


1.45845 


0.35 X 10-* 


5338 


1 . 52989 


0.58 X 10~» 


1.51386 


0.55 X 10"' 


1 . 59463 


0.68 X 10"' 


1.46067 


0.45 X 10-* 


5086 


1.53146 


0.66 X 10"' 


1.51534 


0.63 X 10"' 


1.59644 


0.78 X 10-* 


1.46191 


0.52 X 10-' 


F 4861 


1 . 53303 


0.78 X 10"* 


1.51690 


0.72 X 10"' 


1.59825 


0.89 X 10-' 


1.46318 


. 60 X 10-* 


G' 4340 


1 53790; 1.12 X 10-' 


1.52136 


1.00 X 10"' 


1 . 60367 


1.23 X 10-' 1 1.46690 


0.84 X 10"' 


H 3988 


1 54245] 1.39 X 10~' 


1.52546 


1.26 X 10 * 


1 . 60870 


1.72 X 10-' 1.47030 


1 . 12 X 10"' 



466 



PHYSICAL OPTICS 



length, a curve like one of those in Fig. 23Z? is obtained. The curves 
found for prisms of different optical materials will differ in detail but 
will all have the same general shape. These curves are representative 



1.70 



1.60 




150 



1.40 



2,000 



10,000A 



4,000 6,000 8,000 

Wove length — \— *- 

Fig. 23jB. Dispersion curves for several different materials commonly used for lenses 
and prisms. 

of normal dispersion, for which the following important facts are to be 
noted : 

1. The index of refraction increases as the wavelength decreases. 

2. The rate of increase becomes greater at shorter wavelengths. 

3. For different substances the curve at a given wavelength is usually 
steeper the larger the index of refraction. 

4. The curve for one substance cannot in general be obtained from that 
for another substance by a mere change in the scale of the ordinates. 

The first of these facts agrees with the common observation that in 
refraction by a transparent substance the violet is more deviated than 
the red. The second fact can also be expressed by saying that the 
dispersion increases with decreasing wavelength. This follows because 
the dispersion, dn/dk is the slope of the curve (its negative sign is usually 
disregarded), which increases regularly toward smaller X. An important 
consequence of this behavior of the dispersion is that in the spectrum 



DISPERSION 



467 



formed by a prism the violet end of the spectrum is spread out on a much 
larger scale than the red end. The spectrum is therefore far from being 
a normal spectrum (Sec. 17.6). This will be clear from Fig. 23C, in 
which the spectrum of helium is shown diagrammatically as given by 
flint- and crown-glass prisms and by a grating used under the proper 
conditions to give a normal spectrum. In the prism spectra the wave- 
length scale is compressed toward the red end, as can be seen by com- 
parison with the uniform scale of the normal spectrum. 



(a) 



4,000 



I'll! 



Flint 
glass 



5,000 6,000 7,000 A 



(6) 



4,000 5,000 7,000 



Crown 
glass 



(d 



4,000 



5,000 



6,000 7,000 



Crown 

glass 

(enlarged) 



W) 



Normal 
spectrum 
.(grating) 



4,000 



6.000 



7,000 



5,000 

Wave length — *- 

Fig. 23C. Comparison of the helium spectrum produced by flint- and crown-glass 
prism spectrographs with a normal spectrum. 

The third fact stated above requires that for a substance of highei 
index of refraction, the dispersion dn/dX shall also be greater. Thus, 
comparing (a) and (b) in Fig. 23C, the flint glass has the higher index 
of refraction, and gives a longer spectrum because of its greater dispersion. 
To compare the relative spacing of the lines in (b) with those in (a), the 
spectrum from crown glass has been enlarged, in (c), to have the same 
over-all length between the two lines X3888 and X6678. When this is 
done, it is seen that there is not complete agreement with the lines 
of (a). In fact, the spectra from prisms of different substances will 
never agree exactly in the relative spacing of their spectrum lines. This 
is a consequence of the fourth of the above facts, according to which 
the shape of the dispersion curve is different for every substance. The 
curve for flint glass in Fig. 23B has a greater slope at the violet end, 
relative to that in the red, than does the curve for crown glass. Conse- 
quently, the dispersion of different substances is said to be irrational, 
since there is no simple relation between the different curves. 



468 PHYSICAL OPTICS 

All transparent substances which are not colored show normal disper- 
sion in the visible region. The magnitude of the index of refraction may 
be quite different in various substances, but its change with wavelength 
always shows the characteristics described above. In general, the greater 
the density of the substance the higher its index of refraction and its 
dispersion. For example, flint glass has a density around 2.8, consider- 
ably higher than 2.4 for ordinary crown glass. Water has a smaller n 
and dn/dk, while in a very light substance like air n is practically unity 
and dn/dX very nearly zero. For air n = 1.000276 for red light (Fraun- 
hofer's C line), rising to only 1.000279 for blue light (F line). This rule 
relating density to index of refraction is only a qualitative one, and 
many exceptions are known. For instance, ether has a higher index 
than water (1.36 as compared with 1.33), yet it is less dense, as is shown 
by the fact that ether floats on the surface of water. Similarly, the 
correlation of high dispersion with high index is only rough, and there 
are exceptions to the third rule listed above. Diamond has a density 
of 3.52 and one of the highest known indices of refraction, varying from 
2.4100 for the C line to 2.4354 for the F line. The difference in these 
values, which is a measure of the dispersion, is only 0.0254, whereas a 
dense flint glass may give as much as 0.05 for the same quantity. 

23.3. Cauchy's Equation. The first successful attempt to represent 
the curve of normal dispersion by an equation was made by Cauchy in 
1836. His equation may be written 

w = A + x-* + v 

where A, B, and C are constants which are characteristic of any one 
substance. This equation represents the curves in the visible region, 
such as those shown in Fig. 23i5, with considerable accuracy. To find 
the values of the three constants, it is necessary to know values of n 
for three different X's. Then three equations may be set up which, 
when solved as simultaneous equations, give A, B, and C. For some 
purposes it is sufficiently accurate to include only the first two terms 
and the two constants can be found from values of n at only two X's. 
The two-constant Cauchy equation is, then, 

n = A + ^ 2 (23d) 

from which the dispersion becomes, by differentiation 

This shows that the dispersion varies approximately as the inverse cube 



DISPEESION 



469 



of the wavelength. At 4000 A it will be about eight times as large as 
at 8000 A. The minus sign corresponds to the usual negative slope of 
the dispersion curve. 

The theoretical reasoning on which Cauchy based his equation was 
later shown to be false, so that it is to be considered essentially as an 
empirical equation. Nevertheless it holds very satisfactorily for cases 
of normal dispersion and is a useful equation from a practical standpoint. 
We shall show later that it is a special case of a more complete equation, 
which does have a sound theoretical foundation. 

23.4. Anomalous Dispersion. If measurements of the index of refrac- 
tion of a transparent substance like quartz are extended into the infrared 





Cguchj fx _ \Eguation _ 

i i 

j i 



Visible 
region 

Fig. 232). Anomalous dispersion of a transparent substance like quartz in the infrared. 



Absorption 
band 



region of the spectrum, the dispersion curve begins to show marked 
deviations from the Cauchy equation. The deviation is always of the 
type illustrated in Fig. 23D, where, starting at the point R, the index of 
refraction is seen to fall off more rapidly than required by a Cauchy 
equation that represents the values of n for visible light (between P 
and Q) quite accurately. This equation predicts a very gradual decrease 
of n for large values of X (broken curve), the index approaching the 
limiting value A as X approaches infinity (Eq. 23d). In contrast to this, 
the measured value of n first decreases more and more rapidly as it 
approaches a region in the infrared where light ceases to be transmitted 
at all. This is an absorption band (Sec. 22.3), i.e., a region of selec- 
tive absorption, the position of which is characteristic of the material. 
Within the absorption band, n cannot usually be measured because the 
substance will not transmit radiation of this wavelength. On the long- 
wavelength side of the absorption band the index is found to be very 
high, decreasing at first rapidly and then more slowly as we go farther 



470 



PHYSICAL OPTICS 



beyond the absorption band. Over the range from S to T, the Cauchy 
equation will again represent the data, but with different constants. In 
particular, the constant A will be larger. 

The existence of a large discontinuity in the dispersion curve as it 
crosses an absorption band gives rise to anomalous dispersion. The 
dispersion is anomalous because in this neighborhood the longer wave- 
lengths have a higher value of n and are more refracted than the shorter 
ones. The phenomenon was discovered with certain substances such 
as the dye fuchsine and iodine vapor whose absorption bands fall in 
the visible region. A prism formed of such a substance will deviate 
the red rays more than the violet, giving a spectrum which is very 




f Water 

jr i ^cooling 



Green White 
JS L - 2 



■ ■ ■^ ■ a^^^ffl^^ = 



.^^^ ' Sodium 




Spectroscope 



Fig. 23.E. Experimental arrangement for observing the anomalous dispersion of sodium 
vapor. 

different from that formed by a substance having normal dispersion. 
When it was later discovered that transparent substances like glass and 
quartz possess regions of selective absorption in the infrared and ultra- 
violet, and therefore show anomalous dispersion in these regions, the 
term "anomalous" was seen to be inappropriate. No substance exists 
which does not have selective absorption at some wavelengths, and 
hence the phenomenon, far from being anomalous, is perfectly general. 
The so-called "normal" dispersion is found only when we observe those 
wavelengths which he between two absorption bands, and fairly far 
removed from them. Nevertheless, the term "anomalous dispersion" 
has been retained, although it has little more than historical significance. 
A very striking experiment showing the anomalous dispersion of sodium 
vapor in the neighborhood of the yellow D lines was devised by R. W. 
Wood in 1904. White light when passed through sodium vapor under- 
goes strong selective absorption at these lines, which form a close doublet 
of wavelengths 5890 and 5896 A. At wavelengths far removed from 
these values, the index of refraction is only very slightly greater than 
unity, as we expect for a gas. With sodium vapor of appreciable density, 
the index of refraction in the neighborhood of the D lines passes through 
a region of anomalous dispersion (strictly speaking, two regions very 
close together) of the type shown in Fig. 23D. As the D lines are 
approached from the side of shorter wavelengths, n begins to decrease 
rapidly, becoming much less than unity as we get very close to them. 



DISPERSION 



471 



On the other side, it is at first very high, dropping off rapidly toward 
unity as X increases further. 

To show this in a direct way, Wood made use of the fact that we can 
produce the equivalent of a prism of sodium vapor by vaporizing the 
metal in a partially evacuated tube, if the tube is heated from the bottom. 
The arrangement is shown in Fig. 237?. A number of lumps of metallic 
sodium are placed along the bottom 

of a steel tube provided with water- y p. 

cooled glass windows at the ends and 
an outlet for pumping. White light 
from a narrow horizontal slit Si is 
rendered parallel by the lens Li and 
after passing through the tube, forms 
a horizontal image S[ on the vertical 
slit S 3 of an ordinary prism spectro- 
scope. When the sodium tube is 
cold, S[ will be a sharp, white image, 
illuminating one point of the spec- 
troscope slit, and this will be spread 
out into a narrow horizontal contin- 
uous spectrum in the focal plane of 
the spectroscope camera. If the 
tube is evacuated to about 2 cm 
pressure, and the sodium is heated 
by the row of gas burners, it will va- 
porize slowly, the vapor diffusing 
upward through the residual gas in 
the tube. A density gradient is set 
up, the vapor being densest at the 
bottom and rarest at the top of the 
tube. This is equivalent to a prism 
of vapor, the refracting edge of the 

prism being perpendicular to the plane of the figure, and its thick- 
ness increasing downward. This prism will form an anomalous spec- 
trum on Ss, in which the wavelengths shorter than the yellow (i.e., 
on the green side) are deviated upward, since their n is less than 1, 
and the longer ones (on the orange side) will be deviated downward. As 
a result, we might expect to observe in the spectroscope that the spectrum 
is deviated upward on the green side of the D lines, and downward on 
the red side. (The directions are actually reversed because the spectro- 
scope inverts the image of the slit.) Three actual photographs of the 
resulting spectra with different densities of the vapor are shown in 
Fig. 23F. As a consequence of the inversion mentioned above, the 
photographs form qualitatively a plot of n against X, as in Fig. 23D. In 




Fig. 23F. Anomalous dispersion of sodium 
vapor at three different gas densities. 
(After Cario.) 



472 PHYSICAL OPTICS 

the practical performance of this experiment, several refinements are 
desirable, of which an important one is the introduction of an auxiliary 
diaphragm £ 2 to select that portion of the vapor where the density 
gradient is most uniform.* 

23.5. Sellmeier's Equation. We have seen that the Cauchy equation 
is not capable of representing the dispersion curve in a region of anom- 
alous dispersion. The first success in deriving a formula of more general 
applicability was obtained by postulating a mechanism by which the 
medium could affect the velocity of the light wave. It was assumed that 
the medium contains particles bound by elastic forces, so that they are 
capable of vibrating with a certain definite frequency v . This is the 
so-called natural frequency, i.e., one with which the particles will vibrate 
in the absence of any periodic force, and is identical with the natural 
frequency mentioned in Sec. 22.8 in connection with absorption and 
selective reflection. Passage of the light waves through the medium is 
then assumed to exert a periodic force on the particles, which causes 
them to vibrate. If the frequency v of the light wave does not agree 
with vo, the vibrations will be forced vibrations of relatively small 
amplitude, and of frequency v. As the frequency of the light approaches 
vq, the response of the particles will be greater, and a very large ampli- 
tude will be built up by resonance when v = v exactly. These vibrations 
will in turn react upon the light wave and alter its velocity. A mathe- 
matical investigation of this mechanism was made in 1871 bySellmeier, 
who obtained the equation 

n '- 1+ 5?4rb (23/) 

This equation contains two constants, A and X , the latter being related 
to the natural frequency of the particles by the equation j>oX = c. 
Hence Xo is the wavelength in vacuum corresponding to v . To allow 
for the possibility of the existence of several different natural frequencies, 
the equation can be written with a series of terms, 

i 

in which Xo, Xi, . . . correspond to the possible natural frequencies. 
The constants At are proportional to the number of oscillators capable 
of vibrating with these frequencies. 

Figure 23G is a plot of n against X according to Eq. 2'Ag, assuming two 
natural frequencies. As X approaches X or Xi, n goes to — °° or + <» 

* Further details of the experimental procedure will be found in R. W. Wood, 
"Physical Optics," 3d ed., pp. 492-496, The Macmillan Company, New York, 1934. 



DISPERSION 



473 



on the short-wavelength or long-wavelength side, since the denominator 
of one of the terms in Eq. 2Sg goes to zero. Other important character- 
istics of the curve to be noted are that n approaches unity as X approaches 
zero, and that at X = », n- takes the value 1 + Y A { . 

i 

Sellmeier's equation represents a great improvement over that of 
Cauchy and is in fact identical with that derived from the electromag- 
netic theory (Sec. 23.10) with certain simplifying assumptions. Not only 




Fig. 23G. Theoretical dispersion curves given by Sellmeier's equation for a medium 
havmg two natural frequencies. 

does it take account of anomalous dispersion, but it also gives a more 
accurate representation of n in regions far from absorption bands than 
does a Cauchy equation with the same number of constants. That 
Cauchy 's equation is an approximation to Sellmeier's can be seen by 
writing Eq. 23/ in the form 

A 



n 2 = 1 + 



On expanding by the binomial theorem, we find 

4 /, , X„ 2 , Xo 4 



n 2 = 1 + 



r X 2 ^ x< 



For that part of the dispersion curve where X is considerably greater 
than Xo, the higher powers of X /X will be small and may be neglected. 
This gives 

Xo_ 2 

X 2 



n 2 = 1 + A + A 



474 PHYSICAL OPTICS 

Writing M f or 1 + A, and N for A\ 2 , 

n = (M + 2VA- 2 )* 
Expanding again, 

and neglecting higher oowers of 1/X, 

This is Cauchy's equation as given in Sec. 23.3. 

An instructive experiment to illustrate the origin of dispersion can be 
performed with a simple pendulum, to the bob of which is attached a 
light rubber band. If the end of the rubber band is held in the hand 
and moved to and fro, a periodic force is exerted on the pendulum 
similar to the action of the light wave on one of the oscillators in the 
medium. If the frequency motion of the hand is very high compared 
to the natural frequency of the pendulum, the bob will remain practically 
motionless. This corresponds to a wave of high frequency and short 
wavelength, the velocity of which is practically uninfluenced by the 
presence of the oscillators. In Fig. 23(j it will be seen that n approaches 
unity as X approaches zero, so the velocity becomes the same as in free 
space. 

If, now, the hand is moved with a frequency only slightly greater than 
that of the pendulum, it will be found that the pendulum swings 180° 
out of phase with the motion of the hand. Under these conditions, the 
rubber band is considerably stretched when the displacements of the 
hand and bob are in opposite directions and so exerts its maximum 
force on the hand, tending to pull it back to the central position. This 
corresponds to an increased restoring force on the "aether" which propa- 
gates the wave, and hence to an increase in the velocity of the wave 
(Sec. 11.4). Thus in Fig. 23G, n becomes considerably less than 1 at 
a wavelength slightly less than X . Finally when the frequency of 
motion of the hand is made less than the natural frequency, the pendulum 
will follow the hand, practically in phase with it. In this case the rubber 
band exerts only small forces on the hand, since the displacements of 
the pendulum are in the same direction. The forces are less than if the 
pendulum were at rest, so this corresponds to a decreased restoring force 
on the aether. The velocity of the wave is therefore decreased, and n is 
greater than one, on the long-wavelength side of X . 

The large discontinuity in the dispersion curve at X is thus seen to be 
a consequence of the abrupt change of phase by 180° of the oscillator 
relative to the impressed vibration as the latte